1. Introduction
One of the foundational results connecting syzygies and geometry is Green’s Theorem on linear syzygies of smooth curves.
Theorem 1.1 [Reference GreenGre84a]. Let
$C$
be a smooth curve of genus
$g$
embedded in
${\mathbb P}^n$
via a complete linear series
$|L|$
and
$F$
be the minimal free resolution of the homogeneous coordinate ring of
$C$
. If
$\deg (L) \ge 2g + 1 + p$
for some
$p \ge 0$
, then the embedding
$C \hookrightarrow {\mathbb P}^n$
satisfies the
$N_p$
condition: that is, it is normally generated and
$F_i$
is generated in degree
$i+1$
for
$1 \le i \le p$
.
Let us recall the definitions of the terms in the theorem. The embedding
$C \hookrightarrow {\mathbb P}^n$
is defined to be normally generated (or projectively normal) if the section ring
$\bigoplus _{i \ge 0} H^0(C, L^i)$
is generated in degree 1. Theorem 1.1 gives a common generalization, in the language of syzygies, of two classical results showing that the algebraic presentation of a curve exhibits more rigid structure as its degree grows. Specifically, the
$p = 0$
case of Theorem 1.1 is Castelnuovo’s Theorem, which states that
$C\subseteq {\mathbb P}^r$
is normally generated if
$\deg (L)\geq 2g+1$
[Reference CastelnuovoCas93]; while the
$p=1$
case is a result of Fujita and Saint-Donat stating that
$C$
is cut out by quadrics whenever
$\deg (L) \geq 2g+2$
[Reference FujitaFuj77, Reference Saint-DonatSD72].
Theorem 1.1 helped launch the modern study of the geometry of syzygies. It led to numerous generalizations involving embeddings of surfaces [Reference Agostini, Küronya and LozovanuAKL19, Reference Banagere and HanumanthuBH13, Reference Gallego and PurnaprajnaGP96, Reference Gallego and PurnaprajnaGP99, Reference Küronya and LozovanuKL19, Reference Ng and YeungNY22], smooth higher dimensional varieties [Reference Eisenbud, Green, Hulek and PopescuEGHP05, Reference Ein and LazarsfeldEL93, Reference Gallego and PurnaprajnaGP98, Reference Hering, Schenck and SmithHSS06, Reference Hwang and ToHT13], abelian varieties [Reference ChintapalliChi19, Reference PareschiPar00, Reference Pareschi and PopaPP03, Reference Pareschi and PopaPP04], and Calabi–Yau varieties [Reference NiuNiu19]; see also [Reference Dao and EisenbudDE22, Reference Ein, Niu and ParkENP20]. Theorem 1.1 also led to Green’s Conjecture, which proposes a relationship between the Clifford index of a non-hyperelliptic curve over
$\mathbb {C}$
and the linearity of the free resolution of its coordinate ring with respect to its canonical embedding [Reference GreenGre84b]. This conjecture remains open in general and is a highly active area of research; see e.g. [Reference Aprodu, Farkas, Papadima, Raicu and WeymanAFP+19, Reference Aprodu and VoisinAV03, Reference VoisinVoi02, Reference VoisinVoi05]. For an introduction to the wide circle of ideas on syzygies of curves, see Aprodu and Nagel’s book [Reference Aprodu and NagelAN10], Ein and Lazarsfeld’s survey [Reference Ein and LazarsfeldEL20], Eisenbud’s book [Reference EisenbudEis05], and more [Reference Ein and LazarsfeldEL18, Reference FarkasFar17].
The past two decades have seen a flurry of activity devoted to generalizing work on syzygies to the nonstandard graded setting. For instance, Benson [Reference BensonBen04] generalized Eisenbud–Goto’s Theorem on Castelnuovo–Mumford regularity and linear free resolutions [Reference Eisenbud and GotoEG84, Theorem 1.2(1)] to nonstandard
$\mathbb {Z}$
-graded polynomial rings, leading to breakthroughs in invariant theory due to Symonds [Reference SymondsSym10, Reference SymondsSym11]. Benson’s resulting ‘weighted’ analogue of the Castelnuovo–Mumford regularity was generalized by Maclagan and Smith to multigraded polynomial rings in [Reference Maclagan and SmithMS04], with a view toward applications in toric geometry. This in turn led to much follow-up work on multigraded regularity [Reference Botbol and ChardinBC17, Reference Bruce, Heller and SayrafiBHS21, Reference Bruce, Heller and SayrafiBHS22, Reference Chardin and HolandaCH22, Reference Chardin and NematiCN20], as well as a wide-ranging program on multigraded syzygies [Reference Brown and ErmanBE21, Reference Berkesch, Erman and SmithBES20, Reference Berkesch, Klein, Loper and YangBKLY21, Reference Brown and SayrafiBS22, Reference Busé, Chardin and NematiBCN22, Reference Eisenbud, Erman and SchreyerEES15, Reference Harada, Nowroozi and Van TuylHNVT22, Reference Haiman and SturmfelsHS04, Reference Hering, Schenck and SmithHSS06, Reference Sidman and Van TuylSVT06, Reference YangYan21].
The present work is the first generalization of Green’s Theorem that allows the target of the embedding to be a variety other than projective space, connecting with the literature on nonstandard gradings discussed above, and raising many new questions. For instance, it is far from obvious how to even state an analogue of Theorem 1.1 for curves embedded in weighted projective space. To define weighted
$N_p$
conditions for
$p \ge 1$
, one must ask: What does it mean for a complex of free modules over a nonstandard-
$\mathbb {Z}$
-graded polynomial ring to be linear? To illustrate the subtlety in this question, take a standard graded polynomial ring
$S$
and an
$S$
-module
$M$
generated in degree 0. The minimal free resolution
$F$
of
$M$
is linear if and only if it satisfies the following three equivalent conditions.
-
(1) The differentials in
$F$
are matrices of linear forms. -
(2) The Betti table of
$F$
has exactly one row. -
(3) The degrees of the syzygies grow no faster than those of the Koszul complex.
In the nonstandard
$\mathbb {Z}$
-graded case, each of these yields a distinct analogue of a linear resolution. Furthermore, there is no obvious ‘best’ choice, as each measures something meaningful: (1) leads to strong linearity (Definition 4.1) and the Bernstein–Gel’fand–Gel’fand (BGG) correspondence as in [Reference Brown and ErmanBE22], (2) to weighted regularity and local cohomology as in [Reference BensonBen04], and (3) to Koszul linearity (Definition 4.12) and connections with Koszul cohomology; see § 4 for details on these notions and how they are related. In fact, a central obstacle in our work is the technical challenge of interpolating between these nonequivalent notions of linearity, a challenge that simply is not present in the classical setting.
Phrasing an analogue of Theorem 1.1 also requires weighted versions of a complete linear series and of normal generation. A weighted notion of normal generation is fairly straightforward; see Definition 3.13. But defining a weighted version of a complete linear series turns out to be rather subtle and, as with linearity, there are multiple potential analogues, depending on which aspect of the classical notion one considers. The subtlety arises partly because there is a tension between nondegeneracy and normal generation (see Examples 3.7 and 3.16), and partly because any analogue must depend on data beyond just the line bundle. We propose a log complete series in Definition 3.8 as an analogue of a complete linear series that requires a minimal amount of extra data: a base locus and a degree. These lead to a rich family of examples of embeddings, where the underlying weighted spaces are fairly simple, involving just two distinct degrees.
For our weighted
$N_p$
conditions, we use condition (3) from above.
Definition 1.2. Let
$S$
be the
$\mathbb Z$
-graded polynomial ring corresponding to a weighted projective space
${\mathbb P}(W)$
. Write
$w^i$
for the maximal degree of an
$i$
th syzygy of the residue field. Let
$Z \subseteq {\mathbb P}(W)$
be a variety and
$F=\left [F_0\gets F_1 \gets \cdots \right ]$
the minimal
$S$
-free resolution of its coordinate ring. We say
$Z\subseteq \mathbb {\mathbb P}(W)$
satisfies the weighted
$N_p$
condition if it is normally generated (Definition 3.13), and
$F_i$
is generated in degree
${\leq} w^{i+1}$
for all
$i=1,2, \ldots , p$
(i.e. the complex
$[F_0 \leftarrow \cdots \leftarrow F_p]$
is Koszul 1-linear, in the sense of Definition 4.12).
In the standard grading, we have
$w^{i+1} = i+1$
, so our definition extends Green’s; e.g. if the variables of
$S=k[x_1,x_2,x_3]$
have degrees
$1,2$
, and
$5$
then
$w^1=5, w^2=7$
, and
$w^3=8$
.
We now turn to our main results. We establish the following standing hypotheses.
Setup 1.3. Let
$C$
be a smooth curve of genus
$g$
,
$L$
a line bundle on
$C$
,
$D$
an effective divisor on
$C$
, and
$d\geq 2$
. Assume that
$\deg (L \otimes {\mathcal {O}}(-D)) \ge 2g + 1$
. Let
$W$
be the log complete series of type
$(D,d)$
for
$L$
(see Definition 3.8),
$S = k[x_0, \ldots , x_n]$
the (nonstandard
$\mathbb {Z}$
-graded) coordinate ring of
${\mathbb P}(W)$
, and
$I_C \subseteq S$
the defining ideal of the induced embedding
$C\subseteq {\mathbb P}(W)$
.
Our first result is a weighted generalization of Castelnuovo’s theorem, i.e. the
$p=0$
case of Green’s theorem; see also [Reference Green and LazarsfeldGL86, Reference MattuckMat61, Reference MumfordMum70].
Theorem 1.4.
Under Setup 1.3
, the log complete series
$W$
is normally generated.
The key to Theorem 1.4 is using a suitable generalization of the notion of a complete linear series (Definition 3.8), as many embeddings of curves into weighted projective spaces simply fail to enjoy any reasonable analogue of normal generation; see Example 3.16.
The following generalization of Green’s theorem (Theorem 1.1) is our main result.
Theorem 1.5.
With Setup 1.3: if
$\deg (L \otimes {\mathcal {O}}(-D)) \geq 2g+1+q$
for
$q \ge 0$
then
$C\subseteq {\mathbb P}(W)$
satisfies the weighted
$N_{q+d \cdot \deg (D)}$
condition.
The theorem shows that, even for embeddings into weighted projective spaces, geometric positivity continues to find expression via low degree syzygies, and in a manner that grows uniformly with
$\deg (L)$
. In other words, Green’s fundamental insight from Theorem 1.1 continues to hold for embeddings into weighted projective spaces.
In fact, as the weighted setting has several distinct notions of ‘linearity’, the result even helps bring Green’s result into sharper focus, clarifying that positivity is linked with linearity as defined in relation to the Koszul complex, as opposed to alternate notions of linearity, which are equivalent in the standard grading setting, but not in the weighted setting. In somewhat rough terms, Theorem 1.5 states that, as
$q\to \infty$
, the Betti table of the (weighted) homogeneous coordinate ring of
$C$
‘looks increasingly like the Koszul complex’. The ring
$S$
is standard graded if and only if
$D = 0$
, in which case Theorem 1.5 recovers Green’s Theorem. The reader may find it helpful to look ahead to § 2, where we discuss several detailed examples.
As an immediate consequence of Theorem 1.5, we obtain a generalization of the aforementioned theorem of Fujita and Saint-Donat stating that a curve embedded by a complete linear series of degree
${\geq} 2g+2$
is cut out by quadrics. It is too much to hope that
$C$
will be cut out by
$k$
-linear combinations of products
$x_ix_j$
(see Example 2.1), but this intuition points towards the correct degree bound on the relations.
Corollary 1.6.
With Setup 1.3: if
$\deg (L \otimes {\mathcal {O}}(-D))\geq 2g+2$
then
$I_C$
is defined by equations of degree at most
$2d=\max _{i\ne j} \{ \deg (x_ix_j)\}$
.
In other words, Corollary 1.6 implies that the degrees of the defining equations of
$C$
are bounded by the maximal degree of a syzygy of
$\mathfrak m$
. A number of results in the literature have a similar form to Corollary 1.6, showing that certain relations are generated in degree at most twice the degree of one of the
$k$
-algebra generators, e.g. [Reference Landesman, Ruhm and ZhangLRZ16, Reference Landesman, Ruhm and ZhangLRZ18, Reference SymondsSym11, Reference Voight and Zureick-BrownVZB22].
Our proof of Theorem 1.5 relies on a far more general result relating geometric positivity to low degree syzygies of the section ring
$R:=\bigoplus _{e \geq 0} H^0(C,L^e)$
.
Theorem 1.7.
Let
$C$
be a smooth curve of genus
$g$
,
$L$
a line bundle on
$C$
, and
$f\colon C \to {\mathbb P}(W)$
a closed immersion induced by a weighted series
$W$
associated to
$L$
(see §
3.1
for the definition of a weighted series).
Footnote
1
Assume that
$\deg (L)\geq 2g+1$
and that
$\dim S_1 \gt g$
. Let
$F$
be the minimal free resolution of the section ring
$R$
over the coordinate ring
$S$
of
${\mathbb P}(W)$
. The generators of each
$F_i$
lie in degree
${\leq} w^{i+1}$
for all
$i \leq \dim W - g - 2$
.
Theorem 1.7 highlights that the connection between positivity of an embedding and low degree syzygies is quite robust, as it applies to many situations where normal generation fails. For instance, if we specialize to ordinary projective space, Theorem 1.7 may be applied to obtain low degree syzygies even in cases where a curve is embedded by an incomplete linear series. In fact, many of Green’s results allow for an incomplete linear series, and in the case of an embedding into a standard projective space by an incomplete series, Theorem 1.7 follows from Green’s Vanishing Theorem [Reference GreenGre84a, Theorem 3.a.1]. In the weighted projective case, Theorem 1.7 can be applied quite broadly, as it does not involve a log complete hypothesis.
There is an important distinction between Theorems 1.7 and 1.5: the degree bounds hold for the section ring
$R$
and not for the coordinate ring
$S/I_C$
, respectively. For this reason, Theorem 1.7 yields something like virtual
$N_p$
conditions, where we use ‘virtual’ in the sense of the theory of virtual resolutions introduced in [Reference Berkesch, Erman and SmithBES20], as
$F$
is a virtual resolution of the structure sheaf
${\mathcal O}_C$
. Theorem 1.7 shows that the connection between geometric positivity and low degree syzygies, which was first illuminated by Green, holds in tremendous generality as long as one considers virtual syzygies.
The main theme underlying the technical heart of this paper is the way that, when one passes from a standard to a nonstandard grading, notions of linearity split apart and yet remain subtly intertwined. More specifically, each of the three weighted notions of linearity of free complexes mentioned above, and discussed in detail in § 4, come into play in the following ways.
-
– Koszul linearity is closely linked to geometric positivity; that is, it is the right notion for weighted
$N_p$
conditions. -
– Strong linearity, specifically the multigraded linear syzygy theorem of [Reference Brown and ErmanBE22], is essential to our proof of our key technical result Theorem 1.7.Footnote 2
-
– Weighted regularity plays a crucial role in our proof of Theorem 1.4.
In summary, our main results build on Green’s insight that geometric positivity is expressed algebraically in terms of low degree syzygies, although this requires novel viewpoints on nearly all of the objects involved. Our results provide a proof of concept for the broader idea that the ‘geometry of syzygies’ literature has analogues in the weighted projective setting, and more generally, for embeddings into toric varieties or beyond, bolstering the nascent homological theories for multigraded polynomial rings. Our work also raises a host of new questions: What might play the role of ‘scrolls’ in a weighted projective setting? Is there a weighted analogue of Green’s Conjecture (perhaps for stacky curves)? See § 7 for an array of such questions related to the results in this paper.
Let us now give an overview of the paper. We begin in § 2 with a host of examples illustrating our main results. In § 3 we begin a detailed investigation of closed immersions into weighted projective spaces; we introduce in this section our notion of a ‘log complete series’ and prove a number of foundational results. Section 4 contains a detailed discussion of the various weighted flavors of linear free complexes discussed above. In § 5 we prove Theorem 1.7, the central technical result of the paper. In § 6 we prove the rest of our main results. Finally, in § 7 we outline some follow-up questions raised by this work.
1.1 Notation
Throughout the paper,
$k$
denotes a field and the word ‘variety’ means ‘integral scheme that is separated and a finite type over
$k$
’. Given a vector
${\mathbf {d}} = (d_0, \ldots , d_n)$
of positive integers, we let
${\mathbb P}({\mathbf {d}})$
denote the associated weighted projective space. We always assume that
$d_0\leq d_1 \leq \cdots \leq d_n$
. We often use exponents to indicate the number of weights of a particular degree; for instance, we write
${\mathbb P}(1^3,2^2)$
for
${\mathbb P}(1,1,1,2,2)$
. Given a weighted projective space
${\mathbb P}({\mathbf {d}})$
, we always denote its coordinate ring by
$S$
. That is,
$S$
is the
$\mathbb {Z}$
-graded ring
$k[x_0, \ldots , x_n]$
with
$\deg (x_i) = d_i$
. Alternatively, given a weighted vector space
$W$
, we write
${\mathbb P}(W)$
for the corresponding weighted projective space with coordinate ring
$S=\textrm {Sym}(W)$
. We write
$\mathfrak m$
for the homogeneous maximal ideal of
$S$
.
2. Examples
Before diving into the heart of the paper, we illustrate our main results with some examples. In particular, this section is intended to answer the question: What does a Betti table that satisfies the weighted
$N_p$
condition look like? All computations in
Macaulay2 [GS] of Betti tables throughout this section were performed in characteristic
$0$
.
Theorem 1.5 reduces to Green’s Theorem (Theorem 1.1) when
$\deg (D)=0$
. The simplest new cases are therefore when
$\deg (D)=1$
and
$d=2$
, and so we begin with such examples.
Example 2.1. Let
$C={\mathbb P}^1$
,
$D=[0:1]$
and
$d=2$
. If
$L={\mathcal O}_{{\mathbb P}^1}(2)$
then the corresponding log complete series
$W$
(see Definition 3.8) is
$\langle s^2,st,st^3,t^4\rangle$
. This induces a closed immersion
\begin{equation*} {\mathbb P}^1 \to {\mathbb P}(1^2,2^2) \quad\text { given by } [s:t] \mapsto [s^2:st:st^3:t^4]. \end{equation*}
The defining ideal
$I_C$
for the curve is generated by the
$2\times 2$
minors of
\begin{equation*} \begin{pmatrix} x_0&x_1^2&x_2\\ x_1&x_2&x_3 \end{pmatrix}, \end{equation*}
where
$\deg (x_0)=\deg (x_1)=1$
and
$\deg (x_2)=\deg (x_3)=2$
. Corollary 1.6 implies that
$I$
is generated in degree at most
$4$
, and we can see that this holds and is sharp. A direct computation confirms that
$C\subseteq {\mathbb P}(W)$
is normally generated (see Definition 3.13) and that
$S/I_C$
is a Cohen–Macaulay ring, as predicted by Theorem 1.4. Theorem 1.5 implies that this embedding satisfies the
$N_2$
condition, and we can check this by observing the free resolution
\begin{equation*} F = \left [ S \gets S(-3)^2 \oplus S(-4) \gets S(-5)^2 \gets 0 \right ] \end{equation*}
of
$S/I_C$
. Indeed,
$F_1$
is generated in degrees
${\leq} 4=w^2$
and
$F_2$
is generated in degrees
${\leq} 5=w^3$
and these bounds are sharp.
Example 2.2. Let us continue with the setup of the previous example, but now take
$L={\mathcal O}_{{\mathbb P}^1}(8)$
. The corresponding log complete series
$W$
is spanned by
$s^8, s^{7}t, \ldots ,st^{7}, st^{15}$
and
$t^{16}$
. This weighted series induces a map
${\mathbb P}^1 \to {\mathbb P}(W) = {\mathbb P}(1^8,2^2)$
. In this case, the defining ideal
$I_C$
is given by the
$2\times 2$
minors of the matrix
\begin{equation*} \begin{pmatrix} x_0&x_1&x_2&x_3&x_4&x_5&x_6&x_7^2&x_8\\ x_1&x_2&x_3&x_4&x_5&x_6&x_7&x_8&x_9\\ \end{pmatrix}. \end{equation*}
This ideal is generated in degree at most 4, as predicted by Corollary 1.6. Once again, one can directly check that the embedding is normally generated and that
$S/I_C$
is a Cohen–Macaulay ring. Theorem 1.5 implies that this embedding satisfies the
$N_8$
condition, and one verifies this by inspecting the Betti tableFootnote
3
of
$S/I_C$
.
\begin{equation*} \begin{matrix} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 0: & 1 & . & . & . & . & . & . & . & .\\ 1: & . & 21 & 70 & 105 & 84 & 35 & 6 & . & .\\ 2: & . & 14 & 84 & 210 & 280 & 210 & 84 & 14 & .\\ 3: & . & 1 & 14 & 63 & 140 & 175 & 126 & 49 & 8 \end{matrix} \end{equation*}
Indeed,
$F_i$
is generated in degree
$\leq 3+i=w^{i+1}$
for
$1\leq i \leq 8$
; once again, the bounds imposed by the weighted
$N_p$
conditions are sharp.
Example 2.3. Let
$C$
be the genus
$2$
curve defined by the equation
$z_{2}^{2}-z_{1}^{6}-5z_{0}^{3}z_{1}^{3}-z_{0}^{6}$
in
$\mathbb P(1,1,3)$
. Suppose that
$d=2$
and
$D$
is the single point
$[1:0:1]$
. Let
$L$
be a line bundle of degree 10 on
$C$
. We have
\begin{equation*} \deg (L \otimes {\mathcal {O}}(-D))=9=2g+1+q, \quad \text {with $q=4$.} \end{equation*}
It follows from the Riemann–Roch Theorem that
${\mathbb P}(W) = {\mathbb P}(1^8, 2^2)$
, and so the associated log complete series induces an embedding
$C\subseteq {\mathbb P}(1^8, 2^2)$
. Theorem 1.4 shows that
$C\subseteq {\mathbb P}(1^8,2^2)$
is normally generated and that its coordinate ring is a Cohen–Macaulay ring, which does not seem obvious (at least to these authors). By Theorem 1.5, this embedding satisfies the
$N_{q+d\cdot \deg D}=N_6$
condition. A computation in Macaulay2 yields the following Betti table for
$S/I_C$
.
\begin{equation*} \begin{matrix} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 0: & 1 & . & . & . & . & . & . & . & .\\ 1: & . & 19 & 58 & 75 & 44 & 5 & . & . & .\\ 2: & . & 14 & 80 & 186 & 220 & 136 & 26 & 2 & .\\ 3: & . & 1 & 14 & 61 & 128 & 145 & 98 & 23 & 2\\ 4: & . & . & . & . & . & . & . & 6 & 2 \end{matrix} \end{equation*}
Since
$w^8 = 10$
, we see that the
$7$
th syzygies require a generator of degree
$\gt w^8$
, and thus
$C$
satisfies the weighted
$N_6$
condition, but not the
$N_7$
condition.Footnote
4
The examples so far have been in the case where
$d=2$
and
$\deg D=1$
. We now consider the shape of the Betti table in a slightly different setting.
Example 2.4. Let us consider the case of a genus
$g$
curve
$C$
, where
$d=2$
but now
$\deg D=2$
. Assume that
$\deg (L \otimes {\mathcal {O}}(-D))= 2g+1+q$
for some
$q \ge 0$
. By the Riemann–Roch Theorem, we have
$\dim W_1 = g+q+2$
and
$\dim W_2=4$
, and so
$C \subseteq {\mathbb P}(1^{{g+q+2}},2^4)$
. Theorem 1.5 implies that the minimal free resolution of
$S/I_C$
satisfies the
$N_{q+4}$
condition. Since there are now four variables of degree
$\gt 1$
, the shape of the Betti table is more complicated. Specifically, we have
$w^2=4, w^3 = 6, w^4=8$
, and
$w^{i+1}=w^{i}+1$
for
$i \ge 4$
. This implies that the Betti table of the curve has the following shape, where a symbol
$*$
indicates a potentially nonzero entry.
\begin{equation*} \begin{array}{lccccccccccc} \displaystyle & 0 & 1 & 2 & 3 & 4 & 5 &\ldots &q+4&q+5&\ldots &q+g+4 \\ \displaystyle 0: & * & . & . & . & . & . & \ldots &.&.&\ldots &.\\ \displaystyle 1: & . & * & * & * & * & * & \ldots &*&*&\ldots &*\\ \displaystyle 2: & . & * & * & * & * & * & \ldots &*&*&\ldots &*\\ \displaystyle 3: & . & * & * & * & * & * & \ldots &*&*&\ldots &*\\ \displaystyle 4: & . & .& * & * & * & * & \ldots &*&*&\ldots &*\\ \displaystyle 5: & . & .& . & * & * & * & \ldots &*&*&\ldots &*\\ \displaystyle 6: & . & . & . & . & . & . & \ldots & .&*&\ldots &* \end{array} \end{equation*}
By combining Remark 4.8 and Corollary 6.6, we see that the
$k$
th row of the Betti table vanishes for
$k \gt 6$
. The key moment for the weighted
$N_{q+4}$
condition is in column
$q+5$
, where Theorem 1.5 no longer guarantees that
$F_{q+5}$
is generated in degree
$\leq w^{q+6} =q+10$
.
Remark 2.5. For
$C={\mathbb P}^1$
, the following minor variants of Example 2.1 both lead to non-Cohen–Macaulay examples:
$W= \langle s^2,st,st^3,t^4,t^6\rangle$
and
$W' = \langle s^2,st,st^5,t^6 \rangle$
. This underscores the challenge in finding an appropriate weighted analogue of a complete linear series.
3. Closed immersions into weighted projective spaces
We now begin to lay the technical foundation for this paper, starting with a study of closed immersions into a weighted projective space.
3.1 Weighted series
Let
$Z$
be a variety and
$L$
a line bundle on
$Z$
. A weighted series is a finite-dimensional,
$\mathbb {Z}$
-graded
$k$
-subspace
$W \subseteq \bigoplus _{i \in {\mathbb {Z}}} H^0(Z,L^i)$
. Choosing a basis
$s_0, \ldots , s_n$
of
$W$
, where
$s_i \in W_{d_i} \subseteq H^0(Z, L^{d_i})$
, induces a rational map
${\varphi }_W\colon Z\dashrightarrow {\mathbb P}({\mathbf {d}})$
in exactly the same way as in the case of an ordinary projective space. When the intersection of the zero loci of the
$s_i$
is empty,
${\varphi }_W$
is a well-defined morphism. Let
$S = k[x_0, \ldots , x_n]$
be the
$\mathbb {Z}$
-graded coordinate ring of
${\mathbb P}({\mathbf {d}})$
. We will only be interested in the case where
${\varphi }_W$
is a closed immersion; we describe sufficient conditions for this in Proposition 3.2 below. In this case, let
$I_Z \subseteq S$
denote the homogeneous prime ideal corresponding to the embedding of
$Z$
in
${\mathbb P}({\mathbf {d}})$
. The homogeneous coordinate ring of
${\varphi }_W$
is the
$\mathbb {Z}$
-graded ring
$S / I_Z$
.
Remark 3.1. Before embarking on the results in this section, we highlight some key differences between the behavior of sheaves on weighted and ordinary projective spaces.
-
(1) We have
$\textrm {Pic}({\mathbb P}(d)) = \{{\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(\ell m)\}_{\ell \in {\mathbb {Z}}}$
, where
$m = \textrm {lcm}(d_0, \ldots , d_n)$
[Reference Beltrametti and RobbianoBR86, Theorem 7.1(c)]; in particular, not every sheaf
${\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(i)$
is a line bundle. -
(2) It can happen that
${\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(i) \otimes {\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(j) \ncong {\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(i + j)$
; see e.g. [Reference Beltrametti and RobbianoBR86, pp. 134]. However, it follows from [Reference Beltrametti and RobbianoBR86, Corollary 4A.5(b)] that
${\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(im) \otimes {\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(j) \cong {\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(im + j)$
for all
$i, j \in {\mathbb {Z}}$
. -
(3) Given a graded
$S$
-module
$M$
, it is not always the case that
$\widetilde {M}(j) := \widetilde {M} \otimes {\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(j)$
coincides with
$\widetilde {M(j)}$
. Indeed, taking
$M = S(i)$
, this follows from (2). -
(4) Not every morphism
$Z \to {\mathbb P}({\mathbf {d}})$
arises as
${\varphi }_W$
for some weighted series
$W$
. For instance, take
${\mathbf {d}} = (1,1,2)$
. By (1), every line bundle on
${\mathbb P}({\mathbf {d}})$
is of the form
${\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(2\ell )$
for some
$\ell \in {\mathbb {Z}}$
. In particular,
${\mathcal {O}}_{{\mathbb P}({\mathbf {d}})}(1)$
is not a line bundle, and so there is no line bundle that can induce the map
${\mathbb P}({\mathbf {d}}) \xrightarrow {\textrm {id}} {\mathbb P}({\mathbf {d}})$
.
Proposition 3.2.
Let
$W$
be a weighted series with basis
$s_i \in H^0(Z,L^{d_i})$
for
$0 \le i \le n$
. Assume that there exists
$\ell \gt 0$
such that the map
$S_{\ell } \to H^0(Z, L^{\ell })$
induced by
${\varphi }_W : Z \to {\mathbb P}({\mathbf {d}})$
is surjective and that
$L^{\ell }$
is very ample. The morphism
${\varphi }_W$
is a closed immersion.
Proof.
Let
$f_0, \ldots , f_r$
be a basis of
$H^0(Z,L^{\ell })$
. For
$0 \le i \le r$
, choose
$F_i \in S_{\ell }$
such that
$F_i(s_0, \ldots , s_n)=f_i$
. The linear series determined by the
$F_i$
induces a rational map
$\psi : {\mathbb P}({\mathbf {d}}) \dashrightarrow {\mathbb P}^r$
. Write
$U$
for the domain of definition of
$\psi$
. The image of
${\varphi }_W$
lands in
$U$
since the restriction of the
$F_i$
to
$Z$
is
$f_0, \ldots , f_r$
, which is base-point free. By construction, the composition
$\psi \circ {\varphi }_W$
is the morphism induced by
$|L^{\ell }|$
. Since
$L^{\ell }$
is very ample, the map
$\psi \circ {\varphi }_W$
is a closed immersion, and so
${\varphi }_W$
is a closed immersion into
$U$
[Sta12, 0AGC]; since
$Z$
is proper, it follows that
${\varphi }_W$
is a closed immersion as well.
Remark 3.3. The pathologies described in Remark 3.1 all disappear when one works over the associated weighted projective stack; see [Reference Geraschenko and SatrianoGS15, Section 7] or [Reference PerroniPer08, Theorem 2.6]. However, the stack introduces its own complexities. For instance, the proof of Proposition 3.2 fails: letting
${\mathbb P}_{\text {stack}}(W)$
denote the associated stack and defining
$\tau _W\colon Z \to {\mathbb P}_{\text {stack}}(W)$
in the same way as
${\varphi }_W$
, the composition
$Z\xrightarrow {\tau _W} {\mathbb P}_{\text {stack}}(W)\dashrightarrow {\mathbb P}^r$
being a closed immersion does not imply that
$\tau _W$
is a closed immersion. For a simple counter example, one can let
$Z$
be a point and
$\tau _W$
be any map to a stacky point.
Example 3.4. Let
$Z={\mathbb P}^1$
and
$L={\mathcal O}_{{\mathbb P}^1}(2)$
. The weighted series
$W$
spanned by
$s^2, st \in H^0(Z, L)$
and
$st^3, t^4 \in H^0(Z, L^2)$
induces a map
${\varphi }_W\colon {\mathbb P}^1 \to {\mathbb P}(1,1,2,2)$
given by
$[s:t] \mapsto [s^2:st:st^3:t^4]$
(this is the map in Example 2.1). Applying Proposition 3.2 with
$\ell = 2$
implies that
${\varphi }_W$
is a closed immersion. Indeed, we have a commutative diagram
where the vertical arrow is given by
$[x_0:x_1:x_2:x_3]\mapsto [x_0^2:x_0x_1:x_1^2:x_2:x_3]$
.
Proposition 3.5.
Let
$W$
be a weighted series with basis
$s_i \in H^0(Z,L^{d_i})$
for
$0 \le i \le n$
and
$f\colon S \to \bigoplus _{i \ge 0} H^0(Z, L^i)$
the ring homomorphism given by
$x_i \mapsto s_i$
. If
${\varphi }_W$
is a closed immersion then
$I_Z = \ker (f)$
.
Proof.
The ideal
$I_Z$
is the unique homogeneous prime ideal
$P \subseteq S$
such that
$Z= V(P)$
, where
$V( - )$
is the assignment that sends any homogeneous ideal in
$S$
to its associated subvariety in
${\mathbb P}({\mathbf {d}})$
(see e.g. [Reference Cox, Little and SchenckCLS11, Section 5.2]). Since
$\bigoplus _{i \ge 0} H^0(Z, L^i)$
is a domain,
$\ker (f)$
is prime, and clearly
$V(\ker (f)) = Z$
.
Definition 3.6. A closed immersion
$Z \subseteq {\mathbb P}(\mathbf d)$
is nondegenerate (respectively degenerate) if the defining ideal
$I_Z$
is (respectively is not) contained in
$\mathfrak m^2$
.
Let
$Z\subseteq {\mathbb P}(\mathbf d)$
be a closed immersion with defining ideal
$I_Z\subseteq S$
. Since any basis of
$\mathfrak m/\mathfrak m^2$
can be lifted to give algebra generators for
$S$
, the immersion
$Z \subseteq {\mathbb P}({\mathbf {d}})$
is degenerate if and only if
$I_Z$
contains an element in some minimal generating set for
$\mathfrak m$
. For instance, by examining their defining ideals, one can see that the curves in Examples 2.1 and 2.2 are both nondegenerate.
Example 3.7. Suppose that
$Z={\mathbb P}^1$
,
$L = {\mathcal {O}}(1)$
, and
$W = H^0({\mathbb P}^1, {\mathcal {O}}(1)) \oplus H^0({\mathbb P}^1, {\mathcal {O}}(2))$
. This is a fairly naive way to generalize a complete linear series, as we have simply taken all sections of degrees
$1$
and
$2$
. The map
${\varphi }_W\colon {\mathbb P}^1 \to {\mathbb P}(1^2,2^3)$
given by
$[s:t]\mapsto [s:t:s^2:st:t^2]$
is a closed immersion, by Proposition 3.2. In this case,
$Z \subseteq {\mathbb P}(1^2, 2^3)$
is degenerate since
$I_Z$
contains
$z_2-z_0^2$
,
$z_3-z_0z_1$
, and
$z_4-z_1^2$
.
3.2 Log complete series
We now ask: What is a weighted projective analogue of a complete linear series? Before we state our proposed definition, we fix the following notation: given a divisor
$L$
and an effective divisor
$D$
on a variety
$Z$
, we write
$H^0(Z,L)_D$
for the subspace of sections that vanish along
$D$
.
Definition 3.8. Let
$Z$
be a smooth projective variety,
$L$
a line bundle on
$Z$
,
$D$
an effective divisor on
$Z$
, and
$d\geq 2$
. A weighted series
$W$
is a log complete series of type
$(D,d)$
if
$W_1 = H^0(Z,L)_D$
,
$ H^0(Z,L^d) = W_d \oplus {\textrm {im}}(\textrm {Sym}_d(W_1) \to H^0(Z, L^d))$
, and
$W_i=0$
for
$i\ne 1,d$
.
Example 3.9. Let us revisit Example 2.1, where
$C={\mathbb P}^1$
,
$D=[0:1]$
, and
$d=2$
. In this case,
$W_1 = \langle s^2:st\rangle$
are the sections of
$L$
vanishing at
$D$
; and
$W_2 = \langle st^3:t^4\rangle$
.
Remark 3.10. While a log complete series provides a strong analogue of a complete linear series in the weighted setting, we do not claim that this is a comprehensive analogue. In fact, one could easily imagine minor variants of our setup that would be generated in 3 or more distinct degrees. As with analogues of linear resolutions, we expect that there are distinct analogues of a complete linear series that lead in different directions. We restrict attention to log complete series because they strike a good balance. On one hand, they are sufficiently rich to allow for a wide range of new applications and for our overarching goal of investigating the extent to which Green’s results hold in nonstandard graded settings. On the other hand, they yield embeddings into fairly simple weighted projective spaces of the form
${\mathbb P}(1^a,d^b)$
, thus avoiding some of the pathologies of arbitrary weighted spaces.
When
$D = 0$
, Definition 3.8 recovers the usual notion of a complete linear series. A log complete series
$W$
of type
$(D, d)$
is unique up to isomorphism of graded vector spaces; we will therefore refer to the log complete series of type
$(D, d)$
. Observe that, when
$L^d$
is base-point free, the intersection of the zero loci of the sections in
$W$
is empty, so that
$W$
induces a well-defined morphism
${\varphi }_W\colon Z \to {\mathbb P}({\mathbf {d}})$
.
Lemma 3.11.
Let
$Z$
be a curve,
$L$
a line bundle on
$Z$
, and
$D$
an effective divisor on
$Z$
. There is an isomorphism
$H^0(Z, L^a)_{bD} \cong H^0(Z, L^a \otimes {\mathcal {O}}(-bD))$
for all
$a, b \in {\mathbb {Z}}$
.
Proof.
Since
${\mathcal {O}}_{bD}$
is the structure sheaf of a zero-dimensional scheme,
$L \otimes {\mathcal {O}}_{bD} \cong L$
. Twisting the short exact sequence
$0 \to {\mathcal {O}}(-bD) \to {\mathcal {O}} \to {\mathcal {O}}_{bD} \to 0$
by
$L$
, we therefore arrive at the short exact sequence
$ 0\to L^a \otimes {\mathcal {O}}(-bD)\to L^a \to {\mathcal {O}}_{bD} \to 0.$
The long exact sequence in cohomology yields
$ H^0(Z, L^a)_{bD} := \ker (H^0(Z, L^a) \to H^0(Z, {\mathcal {O}}_{bD})) \cong H^0(Z,L^a \otimes {\mathcal {O}}(-bD) ).$
Proposition 3.12.
Let
$Z,L, D$
, and
$d$
be as in Definition 3.8 and
$W$
the log complete series of type
$(D, d)$
. Assume that
$L^d$
is very ample. The following assertions hold.
-
(1) The canonical map
$S_{d} \to H^0(Z,L^d)$
is surjective. -
(2) The induced map
${\varphi }_W : Z \to {\mathbb P}({\mathbf {d}})$
is a nondegenerate closed immersion. -
(3) The log complete series
$W$
is maximal in the following sense: any weighted series concen-trated in degrees 1 and
$d$
that properly contains
$W$
is degenerate.
Proof.
Part (1) follows from the definition of a log complete series. We may apply Proposition 3.2, with
$\ell = d$
, to conclude that
${\varphi }_W$
is a closed immersion. Nondegeneracy holds since the image of the map
$W_1^{\otimes d} \to H^0(Z, L^d)$
intersects
$W_d$
trivially. This proves (2). If we were to add a section of
$\textrm {Sym}_d(W_1)$
to
$W_d$
then it would be in the image of the map
$W_1^{\otimes d} \to H^0(Z, L^d)$
, forcing degeneracy. Similarly, if we were to add a section
$s$
of
$H^0(Z, L)$
to
$W_1$
, the image of
$W_1^{\otimes d} \to H^0(Z, L^d)$
would intersect
$W_d$
nontrivially; this gives (3).
For explicit examples of log complete series, see Examples 2.1 and 2.2 above.
3.3 A weighted analogue of normal generation
Classically, a closed immersion of a variety
$Z$
in
${\mathbb P}^n$
is projectively normal if the coordinate ring
$S/I_Z$
of the immersion is integrally closed [Reference HartshorneHar77, Example I.3.18]. If
$Z$
is normal, and the closed immersion is induced by the line bundle
$L$
, then the integral closure of
$S/I_Z$
is the section ring
$\bigoplus _{i \in {\mathbb {Z}}} H^0(Z, L^i)$
, and so the immersion is projectively normal if and only if the canonical map
$S \to \bigoplus _{i \in {\mathbb {Z}}} H^0(Z, L^i)$
is surjective. In this case, the line bundle
$L$
is said to be normally generated [Reference MumfordMum70]. Since we have a short exact sequence
\begin{equation*} 0 \to S / I_Z \hookrightarrow \bigoplus _{i \in {\mathbb {Z}}} H^0(Z, L^i) \to H^1_{\mathfrak {m}}(S/I_Z) \to 0 \end{equation*}
of graded
$S$
modules, one concludes that
$L$
is normally generated if and only if
$H^1_{\mathfrak {m}}(S/I_Z) = 0$
. With this in mind, we make the following definition.
Definition 3.13. Let
$Z$
be a variety and
$Z\subseteq {\mathbb P}(\mathbf {d})$
a closed immersion defined by a weighted series
$W$
. We say
$W$
is normally generated if
$H^1_{\mathfrak m} (S/I_Z)=0$
.
Remark 3.14. Let
$Z \subseteq {\mathbb P}({\mathbf {d}})$
be a closed immersion induced by a weighted series
$W$
. The following hold.
-
(1) The weighted series
$W$
is normally generated if and only if the depth of the
$S$
-module
$S/I_Z$
is at least 2. In particular, if
$Z$
is a smooth curve then
$W$
is normally generated if and only if
$S/I_Z$
is a Cohen–Macaulay ring. -
(2) Let
$T = S/I_Z$
. Since
$I_Z$
is prime,
$H^0_{\mathfrak {m}}(T) = 0$
, and so, by [Reference EisenbudEis95, Theorem A4.1], we have a short exact sequence of graded
$S$
modulesThus,
\begin{equation*} 0 \to T\hookrightarrow \bigoplus _{i \in {\mathbb {Z}}} H^0(Z, \widetilde {T(i)}) \to H^1_{\mathfrak {m}}(T) \to 0. \end{equation*}
$W$
is normally generated if and only if the canonical map
$S_i \to H^0(Z, \widetilde {T(i)})$
is surjective for all
$i$
, echoing the classical definition.
Remark 3.15. Unlike the classical case, normal generation of
$W$
is not equivalent to
$S/I_Z$
being integrally closed, even when
$Z$
is normal. For instance, it follows from Theorem 1.4 that the weighted series from Example 2.1 is normally generated. However, using the notation of that example, the ring
$S / I_C = k[s^2, st, st^3, t^4] \subseteq k[s,t]$
is not integrally closed. Indeed,
$t^2 = st^3 / st$
is in the field of fractions of
$S/I_C$
but not in
$S/I_C$
, and it is a root of the polynomial
$z^2 - t^4 \in (S/I_C)[z]$
.
Example 3.16. As mentioned in the introduction, many weighted series fail to be normally generated. For instance, take a weighted series
$W$
such that
$W_1$
is a base-point free, incomplete linear series that yields an embedding
$Z\to {\mathbb P}(W_1)$
. We have
$H^1_{\mathfrak m}(S/I_Z)_1\ne 0$
, and thus,
$W$
fails to be normally generated. We thus see that, for a very positive linear series to have any hope of normal generation, adding a base locus to
$W_1$
is necessary; this observation was a key motivation for our definition of a log complete series.
4. Linearity of free resolutions in the weighted setting
There are multiple ways to extend the definition of a linear free resolution to the weighted setting, each with its advantages and disadvantages. We consider three such notions.
-
(1) Perhaps the most obvious definition of linearity in the weighted setting is strong linearity, which requires all differentials in the resolution to be expressible as
$k$
-linear combinations of the variables; see Definition 4.1 below. This notion was defined and studied in our previous paper [Reference Brown and ErmanBE22], and it is closely related to the multigraded generalization of the BGG correspondence [Reference Hawwa, Hoffman and WangHHW12]. -
(2) We often find that strong linearity is too restrictive for our purposes. There is a weaker, and more well-known, notion of linearity based on a weighted analogue of the Castelnuovo–Mumford regularity and arising from invariant theory [Reference BensonBen04, Reference SymondsSym11], which we call weighted regularity. It is determined by the number of rows in the Betti table of the resolution; see Definition 4.7 for details.
-
(3) Weighted regularity, however, is too weak of a condition for us; we therefore introduce in this paper an intermediate notion between (1) and (2) called Koszul linearity (Definition 4.12). Roughly speaking, a free resolution is Koszul linear if its Betti numbers grow no faster than those of the Koszul complex. Our definition of weighted
$N_p$
conditions (Definition 1.2) is based on Koszul linearity.
Each of (1)–(3) will be used in the proofs of our main results. In the standard graded case, each notion gives an alternative but equivalent way to view linear resolutions; Example 4.16 below illustrates how these notions diverge in the general weighted case. To briefly explain, while weighted regularity only depends on the number of rows in the Betti table of the resolution, Koszul linearity involves more granular information about the Betti numbers. Moreover, strong linearity cannot be detected from the Betti numbers of the resolution at all, as one can see from Example 4.2 below. See also [Reference Brown and ErmanBE23], which explores the relationship between these notions in greater detail.
4.1 Strong linearity
Our most restrictive notion of linearity for nonstandard graded free resolutions is the following.
Definition 4.1 [Reference Brown and ErmanBE22, Definition 1.1]. A complex
$F$
of graded free
$S$
modules is strongly linear if there exists a choice of basis of
$F$
with respect to which its differentials may be represented by matrices whose entries are
$k$
-linear combinations of the variables.
In the nonstandard graded setting, strong linearity of a free complex
$F$
cannot be detected by the degrees of its generators, as the following simple example illustrates.
Example 4.2. Suppose that
$S=k[x_0,x_1]$
, where the variables have degrees
$1$
and
$2$
. Consider the complexes
$S \overset {x_0^2}{\longleftarrow } S(-2)$
and
$S \overset {x_1}{\longleftarrow } S(-2)$
; only the second complex is strongly linear.
The main goal of our previous paper [Reference Brown and ErmanBE22] was to establish a theory of linear strands of free resolutions in the nonstandard graded context, culminating in a generalization of Green’s linear syzygy theorem [Reference GreenGre99]: that circle of ideas will play a key role in this paper. Before we recall the details, we briefly discuss some background on (a weighted analogue of) the BGG correspondence. We refer the reader to [Reference Brown and ErmanBE21, Section 2.2] for a detailed introduction to the multigraded BGG correspondence, following the work of [Reference Hawwa, Hoffman and WangHHW12].
4.1.1 The weighted BGG correspondence
Let
$E = \bigwedge _k(e_0, \ldots , e_n)$
be an exterior algebra, equipped with the
${\mathbb {Z}}^2$
grading given by
$\deg (e_i) = (-\deg (x_i); -1)$
. Denote by
$\textrm {Com}(S)$
the category of complexes of graded
$S$
modules and
${\textrm {DM}}(E)$
the category of differential
$E$
modules, i.e.
${\mathbb {Z}}^2$
-graded
$E$
-modules
$D$
equipped with a degree
$(0; -1)$
endomorphism that squares to 0. The weighted BGG correspondence is an adjunction
\begin{equation*} {\mathbf {L}} : {\textrm {DM}}(E) \leftrightarrows \textrm {Com}(S) : {\mathbf {R}} \end{equation*}
that induces an equivalence on derived categories. We will only be concerned in this paper with the functor
$\mathbf {L}$
applied to
$E$
modules: if
$N$
is a
${\mathbb {Z}}^2$
-graded
$E$
module, the complex
${\mathbf {L}}(N)$
has terms and differential given by
\begin{equation*} {\mathbf {L}}(N)_j = \bigoplus _{a \in {\mathbb {Z}}} S(-a) \otimes _k N_{(a;j)} \quad \text {and} \quad s \otimes n \mapsto \sum _{i = 0}^n x_is \otimes e_in. \end{equation*}
The complex
${\mathbf {L}}(N)$
is strongly linear, and in fact, every strongly linear complex of
$\mathbb {Z}$
-graded
$S$
modules is of the form
${\mathbf {L}}(N)$
for some
$E$
-module
$N$
[Reference Brown and ErmanBE22].
4.1.2 Strongly linear strands
Definition 4.3 [Reference Brown and ErmanBE22]. Let
$M$
be a graded
$S$
module such that there exists
$a \in {\mathbb {Z}}$
with
$M_a \ne 0$
and
$M_{\lt a} = 0$
. We set
$E^* = {\textrm {Hom}}_k(E, k)$
, considered as an
$E$
module via contraction. The strongly linear strand of the minimal free resolution of
$M$
is
${\mathbf {L}}(K)$
, where
$\mathbf {L}$
is the BGG functor defined above, and
\begin{equation*} K = \ker \biggl ( M_a \otimes _k E^*(-a;0) \xrightarrow {\sum _{i = 0}^n x_i \otimes e_i} \bigoplus _{i = 0}^n M_{a + d_i} \otimes _k E^*(-a - d_i ; -1)\biggr). \end{equation*}
In the standard graded case, Definition 4.3 recovers the classical notion of the linear strand of a free resolution [Reference EisenbudEis05, Corollary 7.11]. When
$M$
is generated in a single degree, the strongly linear strand of the minimal free resolution
$F$
of
$M$
may be alternatively defined as follows. it is the unique maximal strongly linear subcomplex
$F'$
of
$F$
such that
$F'$
is a summand (as an
$S$
module, but not necessarily as a complex) of
$F$
[Reference Brown and ErmanBE22].
A main result of [Reference Brown and ErmanBE22] is a multigraded generalization of Green’s Linear Syzygy Theorem [Reference GreenGre99]. We recall the statement of this theorem in the nonstandard
$\mathbb {Z}$
-graded case.
Theorem 4.4 [Reference Brown and ErmanBE22, Theorem 6.2]. Let
$M$
be a finitely generated
$\mathbb {Z}$
-graded
$S$
module and
$F$
its minimal free resolution. Suppose that
$M_0 \ne 0$
, and
$M_i = 0$
for
$i \lt 0$
. The length of the strongly linear strand of
$F$
is at most
$\max \{\dim M_0 - 1, \dim R_0(M)\}$
, where
$R_0(M)$
is the variety of rank one linear syzygies of
$M$
, i.e.
\begin{equation*} R_0(M) = \{w \otimes m \in W \otimes _{\textbf k} M_0 \text { : } wm = 0 \text { in } M\}. \end{equation*}
The following geometric consequence of Theorem 4.4 plays a crucial role in all of our main results. It extends to weighted projective spaces a result originally proven by Green [Reference GreenGre84a] over projective space; see also [Reference EisenbudEis05, Corollary 7.4].
Theorem 4.5.
Let
$Z$
be a variety,
$L$
a line bundle on
$Z$
, and
$W$
a weighted series associated to
$L$
such that the associated map
${\varphi }_W : Z \to {\mathbb P}({\mathbf {d}})$
is a nondegenerate closed embedding. Let
$V$
be a vector bundle on
$Z$
and
$M$
the
$S$
-module
$\bigoplus _{i \in {\mathbb {Z}}} H^0(Z, V \otimes L^i)$
. Assume that
$M_0 \ne 0$
, and
$M_i = 0$
for
$i \lt 0$
. The strongly linear strand of the minimal
$S$
-free resolution of
$M$
has length at most
$\dim {M_0} - 1$
.
While Theorem 4.5 follows directly from ideas in our previous paper [Reference Brown and ErmanBE22] (cf. [Reference Brown and ErmanBE22, Corollary 1.5]), we include a detailed proof here.
Proof.
This follows from essentially the same argument as in [Reference EisenbudEis05, Corollary 7.4] (see also the proof of [Reference Brown and ErmanBE22, Corollary 1.5]). Let
$m \in M_0$
and
$w \in W$
; recall that
$W \subseteq S$
is the
$\textbf k$
-vector subspace of
$S$
generated by the variables. Note that
$m \otimes w \in R_0(M)$
if and only if
$m \otimes w_i \in R_0(M)$
for all homogeneous components
$w_i$
of
$w$
. Assume that
$m \otimes w \in R_0(M)$
and that
$w$
is homogeneous; by Theorem 4.4, it suffices to show that this syzygy is trivial, i.e. either
$m=0$
or
$w=0$
. Suppose that
$m\ne 0$
, and let
$Q$
be a maximal ideal of
$S$
such that the image
$m_Q$
of
$m$
in the localization
$M_Q$
is nonzero. Let
$I_Z$
be the defining ideal of
$Z$
in
${\mathbb P}(W)$
; since
$Z$
is integral,
$I_Z$
is prime. Let
$w_Q$
denote the image of
$w$
in
$(S/I_Z)_Q$
. Note that
$M_Q$
is a free
$R_Q$
-module, where
$R$
is the ring
$\bigoplus _{i \in {\mathbb {Z}}} H^0(Z, L^i)$
. Since
$R$
is a domain, and the natural map
$S/I_Z \to R$
is injective by Proposition 3.5, the relation
$w_Qm_Q = 0$
forces
$w_Q = 0$
, which implies that
$w \in P$
. By the nondegeneracy of the embedding,
$P$
does not contain a homogeneous linear form; we conclude that
$w = 0$
.
We need one additional result concerning strongly linear strands.
Lemma 4.6.
Let
$0\to M'\to M\to M''\to 0$
be a short exact sequence of
$S$
modules. Assume that
$M'_a$
and
$M_a$
are nonzero, and
$M'_i = M_i = 0$
for
$i \lt a$
. Moreover, assume that
$M''_a = 0$
. There is a natural isomorphism between the strongly linear strands of
$M'$
and
$M$
.
Proof.
We assume, without loss, that
$a = 0$
. Let
$L$
be the
${\mathbb {Z}}^2$
-graded
$E$
-module
$\bigoplus _{i = 0}^n M_{ d_i} \otimes _k E^*(-d_i ; -1)$
, and define
$L'$
and
$L''$
similarly. We have a commutative diagram
of
${\mathbb {Z}}^2$
-graded
$E$
modules, where the rows are exact, and the vertical maps are given by multiplication on the left by
$\sum _{i = 0}^n x_i \otimes e_i$
. Let
$K$
(respectively
$K'$
) denote the kernel of the middle (respectively left-most) vertical map. By the Snake lemma, the natural map
$K' \to K$
is an isomorphism, and hence, the natural map
${\mathbf {L}}(K') \to {\mathbf {L}}(K)$
is as well.
4.2 Weighted regularity
Benson introduced in [Reference BensonBen04] an analogue of the Castelnuovo–Mumford regularity for nonstandard
$\mathbb Z$
-graded polynomial rings, which we call ‘weighted regularity’ to emphasize its connection with weighted projective space.Footnote
5
Definition 4.7. Let
$M$
be a finitely generated graded
$S$
module. For each
$i \ge 0$
, set
\begin{equation*} a_i(M) = \sup \{ j \in {\mathbb {Z}} \text { : } H^i_{\mathfrak m}(M)_j \ne 0\}. \end{equation*}
The weighted regularity of
$M$
is
$\sup \{i \ge 0 \text { : } a_i(M)+i\}$
.
Remark 4.8. By a result of Symonds [Reference SymondsSym11, Proposition 1.2], if
$M$
has weighted regularity
$r$
, and
$F$
is the minimal free resolution of
$M$
, then
$F_j$
is generated in degree at most
$r + j + \sum _{i=0}^n (\deg (x_i)-1)$
. Equivalently, the
$k$
th row of the Betti table of any such module vanishes for
$k \gt r + \sum _{i=0}^n (\deg (x_i)-1)$
.
Example 4.9. Let us revisit the two resolutions from Example 4.2. Recall that
$S=k[x_0,x_1]$
, where the variables have degrees
$1$
and
$2$
. Both
$S/(x_0^2)$
and
$S/(x_1)$
have weighted regularity
$0$
, and their Betti tables are both as follows.
\begin{equation*} \begin{matrix} &0&1\\ 0&1&.\\ 1&.&1 \end{matrix} \end{equation*}
In particular, while
$S/(x_0^2)$
has weighted regularity
$0$
, its minimal free resolution is not strongly linear. By contrast, any module that is generated in degree 0 and has a strongly linear free resolution is weighted 0-regular (see Remark 4.13(1) and Proposition 4.14 below).
Example 4.10. In Corollary 6.6, we prove that, under Setup 1.3, the weighted regularity of
$S/I_C$
is
$2$
if
$g\gt 0$
and
$1$
if
$g=0$
. For instance, consider the genus
$2$
curve from Example 2.3 embedded in
${\mathbb P}(1^8,2^2)$
. Its coordinate ring has weighted regularity
$2$
, and so, by Remark 4.8, the Betti table has
$2 + \sum _{i=0}^9 (\deg (x_i)-1) = 2 = 2+ 2 = 4$
rows.
4.3 Koszul linearity
We fix the following.
Notation 4.11. Let
$w^i$
(respectively
$w_i$
) be the sum of the
$i$
largest (respectively smallest) degrees of the variables: that is,
$w^i := \sum _{j=n-i+1}^n d_j$
and
$w_i := \sum ^{i-1}_{j=0} d_j$
.
If
$K=K_0 \gets K_1 \gets \cdots$
is the Koszul complex on
$x_0, \ldots , x_n$
then
$w_i$
is the smallest degree of a generator of
$K_i$
, and
$w^i$
is the largest such degree.
Definition 4.12. A minimal free complex
$[F_0 \overset {{\varphi }_1}{\leftarrow } F_1 \overset {{\varphi }_2}{\leftarrow }F_2 \cdots ]$
of graded
$S$
modules is Koszul
$a$
-linear if each
$F_i$
is generated in degrees
$\lt w^{i+1} + a$
; by minimal we mean
${\varphi }_i(F_i)\subseteq \mathfrak mF_{i-1}$
. We sometimes abbreviate Koszul
$0$
-linear to simply ‘Koszul linear’.
Remark 4.13. We observe the following.
-
(1) If
$M$
is as in Definition 4.12, and the free resolution of
$F$
is Koszul
$a$
-linear, then it follows from Remark 4.8 that
$M$
is weighted
$a$
-regular. The converse is false; see Example 4.16. -
(2) The weighted
$N_p$
condition from Definition 1.2 is equivalent to normal generation of the weighted series and Koszul 1-linearity of the complex
$[F_0 \gets \cdots \gets F_p]$
.
Of course, the Koszul complex on
$x_0, \ldots , x_n$
is Koszul
$0$
-linear. More generally, we have the following proposition.
Proposition 4.14.
Let
$M$
be a graded
$S$
module that is generated in a single degree
$a$
. If the minimal free resolution
$F$
of
$M$
is strongly linear then it is Koszul
$a$
-linear.
Proof.
Since
$F$
is strongly linear and
$M$
is generated in a single degree,
$F$
is equal to its strongly linear strand
${\mathbf {L}}(K)$
, where
$K$
is as in Definition 4.3. It therefore follows from the definition of
$K$
that
$F$
is a summand (as an
$S$
module, but not as a complex) of a direct sum of copies of
${\mathbf {L}}(E^*(-a;0))$
. Finally, observe that
${\mathbf {L}}(E^*(-a;0))$
is the Koszul complex with the
$0$
th term generated in degree
$a$
; the result immediately follows.
Example 4.15. The converse of Proposition 4.14 is false. Returning Example 4.2, the complex
$S \overset {x_0^2}{\longleftarrow } S(-2)$
is Koszul
$0$
-linear but not strongly linear.
Example 4.16. Let
$C = {\mathbb P}^1$
,
$L = {\mathcal {O}}_C(5)$
, and
$D$
the divisor
$[0:1] + [1:0]$
. The associated log complete seres induces an embedding
${\mathbb P}^1 \subseteq {\mathbb P}(1^4,2^4)$
given by
\begin{equation*} [s:t]\mapsto [s^{4}t:s^{3}t^{2}:s^{2}t^{3}:st^{4}:s^{9}t:s^{10}:st^{9}:t^{10}]. \end{equation*}
The Betti table is as follows.
\begin{equation*} \begin{matrix} & 0 & 1 & 2 & 3 & 4 & 5 & 6\\ \text {total:} & 1 & 21 & 70 & 105 & 84 & 35 & 6\\ 0: & 1 & . & . & . & . & . & .\\ 1: & . & 3 & 2 & . & . & . & .\\ 2: & . & 12 & 24 & 12 & . & . & .\\ 3: & . & 6 & 36 & 54 & 24 & . & .\\ 4: & . & . & 8 & 36 & 48 & 20 & .\\ 5: & . & . & . & 3 & 12 & 15 & 6 \end{matrix} \end{equation*}
From this Betti table, one can check that this resolution is Koszul
$1$
-linear. For instance,
$F_1$
has generators of degree
$\lt 5 = w^{2}+1$
,
$F_2$
has generators of degree
$\lt 7 = w^{3}+1$
, and so on.
The defining ideal
$I_C$
is given by the
$2\times 2$
minors of the matrix
$\left (\begin{smallmatrix} x_{0}&x_{1}&x_{2}&x_{4}&x_{5}&x_{3}^{2}&x_{6}\\ x_{1}&x_{2}&x_{3}&x_{0}^{2}&x_{4}&x_{6}&x_{7}\\ \end{smallmatrix}\right )$
. It follows that the minimal free resolution of
$S/I_C$
is the Eagon–Northcott complex of this matrix. Since this matrix includes the entries
$x_3^2$
and
$x_0^2$
, this minimal free resolution is not strongly linear. Thus, even in the case of a rational curve, strong linearity is too restrictive to capture the linearity of the free resolution of the coordinate ring.
Finally, let us analyze the example from the perspective of weighted regularity. By Remark 4.13(1),
$S/I_C$
is
$1$
-regular; by Remark 4.8, this states precisely that the
$k$
th row of the Betti table vanishes for
$k \gt 5$
. Thus, for instance, the weighted regularity computation would imply that
$F_1$
is generated in degree at most
$6$
. We therefore see that weighted regularity is too weak to fully describe the situation.
5. Proof of Theorem 1.7
We begin by establishing several technical results. The first is a simple calculation.
Lemma 5.1.
Let
$S$
be as in Theorem
1.7
and
$M$
be a finitely generated
$S$
module. Assume that
$M_0\ne 0$
but
$M_i=0$
for
$i\lt 0$
. The following assertions hold.
-
(1) If the Betti number
$\beta _{i, j}(M)$
is nonzero then
$j \ge w_i$
(see Notation
4.11
).
-
(2) Suppose that there is a variable
$x_\ell \in S$
that is a nonzero divisor on
$M$
. Define
If
\begin{equation*} w_i' = \begin{cases} w_i, & i \lt \ell , \\ w_{i+1} - \deg (x_\ell ), & i \ge \ell . \end{cases} \end{equation*}
$\beta _{i, j}(M) \ne 0$
then
$j \ge w_i'$
.
Proof.
If
$K$
is the Koszul complex on the variables
$x_0, \ldots , x_n$
then the minimal degree of an element of
$\textrm {Tor}_i(M,k)=H_i(M\otimes _{S} K)$
is
$w_i$
. This proves (1). For (2), let
$F$
denote the minimal
$S$
-free resolution of
$M$
. Since
$x_\ell$
is a nonzero divisor on
$M$
,
$F / x_\ell F$
is the minimal
$S / (x_\ell )$
-free resolution of
$M / x_\ell M$
. Now apply (1) to the
$S / (x_\ell )$
-module
$M / x_\ell M$
.
The following lemma is an analogue of a well-known result in the standard graded case and is proven in the same way as its classical counterpart.
Lemma 5.2.
Let
$C, L$
,
$R$
,
$S$
,
$W$
, and
$f\colon C \to {\mathbb P}(W)$
be as in Theorem
1.7
. The following hold.
-
(1) The graded
$S$
-module
$R$
has depth 2. In particular,
$R$
is a Cohen–Macaulay
$S$
module and a maximal Cohen–Macaulay
$S / I_C$
module.
-
(2) Let
${\omega }_R = \bigoplus _{i \in {\mathbb {Z}}} H^0(C, {\omega }_C \otimes L^i)$
, and denote by
$|{\mathbf {d}}|$
the sum of the degrees of the variables in
$S$
. We have
$ {Ext}^{n-1}_S(R, S(-|\mathbf {d}|)) \cong {\omega }_R.$
Proof.
We observe that the canonical map
$R \to \bigoplus _{i \in {\mathbb {Z}}} H^0({\mathbb P}(W), \widetilde {R(i)} )$
is an isomorphism, i.e.
$R$
is
$\mathfrak m$
saturated. By [Reference EisenbudEis95, Theorem A4.1], we have an exact sequence
\begin{equation*} 0\to H^0_{\mathfrak m} (R) \to R \xrightarrow {\cong } \bigoplus _{i \in {\mathbb Z}} H^0({\mathbb P}(W), \widetilde {R(i)}) \to H^1_{\mathfrak m} (R)\to 0, \end{equation*}
and isomorphisms
\begin{equation} H^{j+1}_{{\mathfrak m}} (R) \cong \bigoplus _{i \in {\mathbb {Z}}} H^j({\mathbb P}(W), \widetilde {R(i)}) = \bigoplus _{i \in {\mathbb {Z}}} H^j(C, L^i), \end{equation}
for
$j \gt 0$
. In particular, we have
$H^i_{\mathfrak m} (R) = 0$
for
$i = 0, 1$
; that is,
$R$
has depth 2. Part (1) now follows from the observation that
$\dim {S / I_C} = 2$
. As for (2), given a
$\mathbb {Z}$
-graded
$k$
-vector space
$V$
, let
$V^*$
denote its graded dual. We have
$ {Ext}^{n-1}_S(R, S(-|{\mathbf {d}}|)) \cong H^2_{\mathfrak {m}}(R)^* \cong \bigoplus _{i \in {\mathbb {Z}}} H^1(C, L^i)^*\cong {\omega }_R,$
where the first isomorphism follows from local duality, the second from (5.3), and the third from Serre duality.
Next, we need the following strengthening of Theorem 4.5.
Lemma 5.4.
Suppose we are in the setting of Theorem
4.5
, and assume that
$\dim W_1 \gt \dim M_0$
. Let
$F$
be the minimal
$S$
-free resolution of
$M$
. Any summand of
$F_i$
generated in degree
$j$
for some
$j \lt w_{i+1}$
(see Notation
4.11
) lies in the strongly linear strand of
$F$
. In particular, if
$\beta _{i,j}(M)\ne 0$
for some
$j \lt w_{i+1}$
then
$i \le \dim M_0 - 1$
.
In the standard graded case, the first statement in Lemma 5.3 is tautological: it says that, if a summand of
$F_i$
is generated in degree
$i$
then it is in the linear strand. However, in the weighted setting, the strongly linear strand cannot be interpreted in terms of Betti numbers (see, for instance, Example 4.2), and so Lemma 5.3 is not at all obvious in general; indeed, our proof is a bit delicate.
Proof of Lemma 5.4. The second statement follows immediately from the first, by Theorem 4.5. Let
$K$
be the Koszul complex on the variables of
$S$
. We consider classes in
$\textrm {Tor}^S_*(k, M)$
as homology classes in
$K \otimes _S M \cong \bigwedge W \otimes _k M$
, and we fix once and for all an embedding
$\textrm {Tor}^S_*(k, M) \hookrightarrow Z(\bigwedge W \otimes _k M)$
of
${\mathbb {Z}}^2$
-graded
$k$
-vector spaces, where the target denotes the cycles in
$\bigwedge W \otimes _k M$
. In this proof, we identify classes in
$\textrm {Tor}^S_*(k, M)$
with cycles in
$\bigwedge W \otimes _k M$
via this embedding. We may decompose any element
$\sigma \in \bigwedge W \otimes _k M$
as
$\sum _{i\geq 0} \sigma _i$
, where
$\sigma _i \in \bigwedge W \otimes _k M_i$
. Let
$W_{\gt 1} = \bigoplus _{i \gt 1} W_i$
, so that
$\bigwedge W = \bigwedge W_1 \otimes _k \bigwedge W_{\gt 1}$
. We may write any
$\sigma \in \bigwedge W \otimes _k M$
as
$\sum \alpha \otimes \beta \otimes m_{\alpha ,\beta }$
, where the sum ranges over all pairs
$(\alpha , \beta )$
such that
$\alpha$
is an exterior product of basis elements of
$W_1$
, and
$\beta$
is an exterior product of basis elements of
$W_{\gt 1}$
; here, each
$m_{\alpha ,\beta }$
is an element of
$M$
. We call each nonzero
$\alpha \otimes \beta \otimes m_{\alpha ,\beta }$
in this sum a term of
$\sigma$
. It is possible that
$\alpha$
(respectively
$\beta$
) is an empty product of basis elements, in which case
$\alpha$
(respectively
$\beta$
) is
$1 \in \bigwedge ^0 W_1$
(respectively
$1 \in \bigwedge ^0 W_{\gt 1}$
). Given a nonzero element
$\sigma \in \bigwedge W \otimes _k M$
, we define
\begin{equation*} \nu (\sigma ) = \max \bigg\{ m \text { : } \text {a term of $\sigma $ lies in } \bigwedge ^{\dim W_1 - m}W_1 \otimes _k \bigwedge W_{\gt 1}\otimes _k M \bigg\}. \end{equation*}
The function
$\nu$
measures the maximal number of degree
$1$
elements that do not appear in one of the exterior forms
$\alpha$
. For instance, if
$\nu (\sigma )=0$
then, for every term
$\alpha \otimes \beta \otimes m_{\alpha ,\beta }$
of
$\sigma$
,
$\alpha$
is the product of all of the degree
$1$
variables. Let us now prove the following.
Claim.
If
$\sigma$
is a nonzero class in
$\textrm {Tor}_*^S(k, M)$
then
$\nu (\sigma ) \ne 0$
.
Proof of Claim. Indeed, let
$x_i$
be a degree 1 variable,
$\overline {W}$
the quotient of
$W$
by the span of
$x_i$
, and
$\overline {M}$
the corresponding module
$M/(x_i)$
over
$\overline {S} = S/(x_i)$
. Since
$x_i$
is a regular element on
$M$
, the surjection
$\bigwedge W\otimes _k M \twoheadrightarrow \bigwedge \overline {W} \otimes _k \overline {M}$
induces an isomorphism
$\theta : \textrm {Tor}^S_*(k,M) \xrightarrow {\cong } \textrm {Tor}_*^{\overline {S}}(k,\overline {M})$
on homology. Since
$\theta (\sigma ) \ne 0$
,
$\nu (\sigma )$
must be nonzero; this proves the claim.
Now, let
$\sigma$
be a nonzero class in
$\textrm {Tor}^S_i(k, M)_j$
, where
$j \lt w_{i+1}$
. It suffices to show that
$\sigma = \sigma _0$
; this implies that
$\sigma$
lies in the strongly linear strand. Assume, toward a contradiction, that
$\sigma _\ell \ne 0$
for some
$\ell \gt 0$
. Since
$\sigma _\ell \in \bigwedge ^i W \otimes M_\ell$
, we have
$w_i + \ell \le j \lt w_{i+1}$
. Recalling that
$w_{i+1} - w_i = d_{i+1} := \deg (x_{i+1})$
, this implies that
$d_{i+1} \gt \ell \geq 1$
. We conclude that
\begin{equation} i \geq \dim W_1. \end{equation}
There are two cases to consider.
Case 1:
$\nu (\sigma _\ell )\gt 0$
for some
$\ell \gt 0$
. In this case,
$\sigma _\ell$
has some term
$\alpha \otimes \beta \otimes m_{\alpha ,\beta }$
such that
$\alpha$
is not divisible by a degree
$1$
variable; without loss of generality, let us say
$\alpha$
is not divisible by
$x_0$
. It follows that
$\deg (\alpha \otimes \beta ) \geq \deg (x_1x_2\cdots x_i)=w_{i+1}-1$
. Thus,
\begin{equation*} \deg (\sigma _\ell ) = \deg (\alpha \otimes \beta ) + \ell \geq w_{i+1}-1+\ell \ge w_{i+1}. \end{equation*}
This is impossible, since
$\deg (\sigma _\ell ) = \deg (\sigma ) \lt w_{i+1}$
.
Case 2:
$\nu (\sigma _\ell ) = 0$
for all
$\ell \gt 0$
. For every term
$\alpha \otimes \beta \otimes m_{\alpha , {\beta }}$
of
$\sigma _\ell$
for
$\ell \gt 0$
, we have
$\beta \in \bigwedge ^{i-\dim W_1}W_{\gt 1}$
. On the other hand, it follows from the claim above that there must be some term
$\alpha ' \otimes \beta '\otimes m_{\alpha ',\beta '}$
of
$\sigma _0$
such that
$\beta ' \in \bigwedge ^{i-\dim W_1+t}W_{\gt 1}$
for some
$t\gt 0$
; recall that, by (5.5),
$i - \dim W_1 \ge 0$
. Let
$E = \bigwedge W^*$
, and note that
$\bigwedge W \otimes _k M$
is an
$E$
module via the contraction action of
$E$
on
$\bigwedge W$
. We may choose
$f \in \bigwedge ^{i-\dim W_1+1}W_{\gt 1}^* \subseteq E$
such that
$f\sigma _0\ne 0$
; note, however, that
$f\sigma _\ell =0$
for all
$\ell \gt 0$
. Thus,
$f\sigma = f\sigma _0=(f\sigma )_0 \in \bigwedge W \otimes M_0$
. Moreover, since
$\sigma \in \bigwedge W \otimes _k M$
is a cycle,
$f\sigma$
is also a cycle, as the Koszul differential on
$\bigwedge W \otimes _k M$
is
$E$
-linear. Thus, since
$f\sigma = (f\sigma )_0$
, it follows from the definition of the strongly linear strand (Definition 4.3) that
$f\sigma$
determines a summand of the strongly linear strand of
$F$
. But
$f\sigma$
has homological degree
$i- (i-\dim W_1 + 1)= \dim W_1 -1$
, and so
$\dim W_1 -1 \leq \dim M_0-1$
, by Theorem 4.5. This contradicts our assumption that
$\dim W_1 \gt \dim M_0$
.
In the standard graded case, the proof of Green’s Theorem (Theorem 1.1) via the linear syzygy theorem (cf. [Reference EisenbudEis05, Theorem 8.8.1]) makes use of numerous statements about linear strands that rely on degree arguments. These break down in the nonstandard graded situation, and Lemmas 5.1–5.3 act to fill that gap. Thus, with these lemmas in hand, we can now turn to the proof of Theorem 1.7.
Proof of Theorem 1.7. Recall that
$d_0, \ldots , d_n$
are the degrees of the variables
$x_0, \ldots , x_n$
in
$S$
, and we assume that
$d_0 \le d_1 \le \cdots \le d_n$
. As in Lemma 5.2(2), we let
$|{\mathbf {d}}| = \sum _{i = 0}^n d_i$
and
${\omega }_R=\bigoplus _{i \in {\mathbb {Z}}} H^0(C,\omega _C \otimes L^i)$
. We remark, for later use, that
$\dim ({\omega }_R)_0=H^0(C,\omega _C) = g$
. By Lemma 5.2(2), we have
$ {Ext}^{n-1}_S(R, S(-|\mathbf {d}|)) \cong {\omega }_R.$
Letting
$F$
be the minimal
$S$
-free resolution of
$R$
and
$F^\vee = {\textrm {Hom}}_S(F, S)$
, it follows that
$F^\vee (- |\mathbf {d}|)[-n+1]$
is the minimal free resolution of
${\omega }_R$
. In particular, we have
$ \beta _{i,j}(R) = \beta _{n - 1 - i, |\mathbf {d}| - j}({\omega }_R).$
Now, suppose that
$\beta _{i, j}(R) = \beta _{n-1-i, |\mathbf {d}| - j}({\omega }_R) \ne 0$
, and assume that
$j \gt w^{i+1}$
. We now compute
\begin{equation*} |\mathbf {d}| - w_{n-i} = \sum _{j=0}^n d_j - \sum _{j=0}^{n-1-i} d_j = \sum _{j=n-i}^n d_j = w^{i+1} \lt j. \end{equation*}
Rearranging this inequality, we have
$|{\mathbf {d}}| - j \lt w_{n-i}$
. There are now two cases to consider.
Case 1:
$g = 0$
. In this case,
$({\omega }_R)_1 \ne 0$
, and
$({\omega }_R)_i = 0$
for
$i \lt 1$
. Every variable
$x_i \in S$
is a nonzero divisor on
${\omega }_R$
. In particular,
$x_0$
has this property; recall that
$\deg (x_0) = 1$
. Applying Lemma 5.1(2), with
$\ell = 0$
, we arrive at the inequality
$|{\mathbf {d}}| -j\ge w_{n-i}$
, a contradiction. We therefore conclude that if
$\beta _{i, j}(R) \ne 0$
then
$j \le w^{i+1}$
.
Case 2:
$g \gt 0$
. We now have
$({\omega }_R)_0 \ne 0$
, and
$({\omega }_R)_i = 0$
for
$i \lt 0$
. Applying Lemma 5.3 to
${\omega }_R$
implies that
$n - 1 - i \lt \dim ({\omega }_R)_0 = g$
, i.e.
$i \gt n - 1 - g = \dim W - g - 2$
.
Let us illustrate the proofs of both Theorem 1.7 and Lemma 5.3 via an example.
Example 5.6. Suppose we are in the setting of Theorem 1.7, and assume that
$g = 2$
and
${\mathbb P}(W) = {\mathbb P}(1^6,2^4)$
. Let
${\omega }_R$
be as in Lemma 5.2(2). To prove Theorem 1.7 in this example, we must show that the columns of the Betti table of
${\omega }_R$
are bounded above by the dots in the diagram below.Footnote
6
\begin{equation*} \begin{matrix} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 0: & \bullet & \bullet & . & . & . & . & . & . & .\\ 1: & . &. & \bullet & \bullet & \bullet & \bullet &\dagger & . & .\\ 2: & . & . & . &. & . & . & \bullet & . &.\\ 3: & . & . & . & . & . & . & . & \bullet & .\\ 4: & . & . & . & . & . & . & . & . & .\\ 5: & . & . & . & . & . & . & . & . & .\\ 6: & . & . & . & . & . & . & . & . & \bullet \end{matrix} \end{equation*}
For degree reasons alone, entries in the
$0$
th row must lie in the strongly linear strand of the minimal free resolution of
${\omega }_R$
, and the length of that strand is
$\leq g-1=1$
by [Reference Brown and ErmanBE22, Corollary 1.4]. So the first entry that could potentially pose an issue is the one in the position marked by a
$\dagger$
, as we cannot conclude, for purely degree reasons, that such an entry lies in the strongly linear strand. Let us use the argument in the proof of Lemma 5.3 to show this entry must be 0.
We adopt the notation of the proof of Lemma 5.3. Say we have a cycle
$\sigma \in \bigwedge ^6 W \otimes {\omega }_R$
corresponding to a nonzero syzygy in position
$\dagger$
. For degree reasons, we have
$\sigma _i = 0$
for
$i \ne 0, 1$
; and
$\nu (\sigma _1)=0$
. In particular, we have
$\sigma _1 = x_0x_1\cdots x_5 \otimes y$
for some
$y \in ({\omega }_R)_1$
. It follows that, for every
$f\in W_2^*$
, we have
$f\sigma _1=0$
. The claim in the proof of Lemma 5.3 implies that
$\nu (\sigma )\ne 0$
, and thus,
$\sigma _0$
must be nonzero and satisfy
$\nu (\sigma _0)\gt 0$
. In particular, every term of
$\sigma _0$
must involve at least one variable from
$W_{\gt 1}$
. We can thus choose an element
$f\in W_{\gt 1}^*$
such that
$f\sigma _0 \ne 0$
. We therefore have
$f\sigma = f\sigma _0 + f\sigma _1 = f\sigma _0 \ne 0$
, which means
$f\sigma$
corresponds to a summand of the strongly linear strand that lies in the position of the entry marked
$\star$
below.
\begin{equation*} \begin{matrix} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 0: & \bullet & \bullet & . & . & . & \star & . & . & .\\ 1: & . &. & \bullet & \bullet & \bullet & \bullet &\dagger & . & .\\ 2: & . & . & . &. & . & . & \bullet & . &.\\ 3: & . & . & . & . & . & . & . & \bullet & .\\ 4: & . & . & . & . & . & . & . & . & .\\ 5: & . & . & . & . & . & . & . & . & .\\ 6: & . & . & . & . & . & . & . & . & \bullet \end{matrix} \end{equation*}
This is impossible, because the strongly linear strand has length at most
$g-1=1$
.
6. Normal generation and the weighted
$N_p$
results
We use the notation/assumptions in Setup 1.3 throughout this entire section. Recall that
${\varphi }_W\colon C \to {\mathbb P}({\mathbf {d}})$
is a closed embedding, by Proposition 3.12(2). As above, we denote by
$R$
the section ring
$\bigoplus _{i \in {\mathbb {Z}}} H^0(C, L^i)$
, and we write
$H^0(C,L^a)_{bD}$
for the space of sections of
$L^a$
that vanish along the divisor
$bD$
. We begin with several technical results.
Lemma 6.1.
We have
$(S/I_C)_{\ell d} \cong R_{\ell d}$
for all
$\ell \ge 0$
.
Proof.
By Proposition 3.5, we need only show that the ring map
$\alpha \colon S \to \bigoplus _i H^0(C, L^{i})$
given by
$x_i \mapsto s_i$
induces surjections
$\alpha _{\ell d} \colon S_{\ell d} \twoheadrightarrow H^0(C, L^{\ell d})$
for all
$\ell \ge 0$
. By Proposition 3.12(1),
$\alpha _{d}$
is surjective. Let
$V = H^0(C, L^d)$
,
$f_0, \ldots , f_r$
a basis of
$V$
, and
$F_0, \ldots , F_r \in S_d$
elements such that
$\alpha _d(F_i) = f_i$
. Let
$\ell \ge 0$
. By our assumption on
$\deg (L \otimes {\mathcal {O}}(-D))$
, the embedding
$C \hookrightarrow {\mathbb P}(V)$
determined by
$|L^d|$
is normally generated, and so the induced map
$h \colon \textrm {Sym}^\ell (V) \to H^0(C, L^{\ell d})$
is surjective. Let
$s \in H^0(C, L^{\ell d})$
, and choose
$p \in \textrm {Sym}^\ell (V)$
such that
$h(p) = s$
, i.e.
$p(f_0, \ldots , f_r) = s$
. We have
$\alpha _{\ell d}(p(F_0, \ldots , F_r) )= s$
.
Lemma 6.2.
Let
$e \geq 0$
, and write
$e = qd+e'$
for
$0 \le e' \lt d$
. The following assertions hold.
-
(1) The natural map
$ H^0(C,L^{qd}) \otimes H^0(C,L^{e'} \otimes {\mathcal {O}}(-e'D)) \to H^0(C,L^e \otimes {\mathcal {O}}(- e'D))$
is surjective.
-
(2) The image of the injection
$(S/I_C)_e \hookrightarrow H^0(C, L^e)$
is given by the sections that vanish with multiplicity
$\geq e'$
along
$D$
.
Proof.
Part (1) is immediate from [Reference GreenGre84a, Corollary 4.e.4]. As for (2): let
$\iota$
denote the injection
$(S/I_C)_e \hookrightarrow H^0(C, L^e)$
. Because
$C$
is embedded by a log complete series of type
$(D,d)$
, the variables of
$S = k[x_0, \ldots , x_n]$
have degrees
$1$
and
$d$
. Say
$x_0, \ldots , x_r$
are the variables of degree 1. Every element of
$S_e$
, and hence
$(S/I_C)_{e}$
, lies in
$(x_0, \ldots ,x_r)^{e'}$
. It follows that every section in the image of
$g$
vanishes with multiplicity
$\geq e'$
along
$D$
; that is,
${\textrm {im}}(\iota ) \subseteq H^0(C,L^e)_{e'D}$
. By Lemma 3.11, there is a natural isomorphism
$H^0(C,L^e)_{e'D} \cong H^0(C,L^e \otimes {\mathcal {O}}(-e'D))$
. Since
$\deg (L \otimes {\mathcal {O}}(-D))\geq 2g+1$
, the complete linear series on
$L \otimes {\mathcal {O}}(-D)$
induces a normally generated embedding into projective space, i.e. the natural map
$H^0(C,L \otimes {\mathcal {O}}(-D))^{\otimes a} \to H^0(C,L^a \otimes {\mathcal {O}}(-aD))$
is surjective for all
$a\geq 0$
.
We first consider the case where
$e \lt d$
, so that
$e=e'$
and
$q=0$
. We have isomorphisms
$(S/I_C)_1= H^0(C,L)_D \cong H^0(C,L \otimes {\mathcal {O}}(-D))$
and
$H^0(C,L^e \otimes {\mathcal {O}}(-eD))\cong H^0(C,L^e)_{eD}$
. Combining these identifications with the surjection
$H^0(C,L \otimes {\mathcal {O}}(-D))^{\otimes e} \twoheadrightarrow H^0(C,L^e \otimes {\mathcal {O}}(-eD))$
yields a surjection
$\pi \colon (S/I_C)_1^{\otimes e}\twoheadrightarrow H^0(C,L^e)_{eD}$
. We have a commutative diagram
where the vertical map is the inclusion, and the left-most horizontal map is given by multiplication. This proves (2) when
$e\lt d$
. Finally, suppose that
$e\geq d$
. By Lemma 6.5, we have
$(S/I_C)_{\ell d} \cong R_{\ell d} = H^0(C,L^{\ell d})$
for all
$\ell \geq 0$
, and we have shown above that
$(S/I_C)_{e'} \cong H^0(C,L^{e'} \otimes {\mathcal {O}}(-e'D))$
. Part (1) yields a surjection
\begin{equation*} H^0(C,L^{qd}) \otimes H^0(C,L^{e'} \otimes {\mathcal {O}}(-e'D)) \twoheadrightarrow H^0(C,L^e \otimes {\mathcal {O}}(-e'D))\cong H^0(C,L^e)_{e'D}. \end{equation*}
Combining these observations, we see that there is a surjection
$\pi : (S/I_C)_{qd}\otimes (S/I_C)_{e'} \twoheadrightarrow H^0(C,L^e)_{e'D}$
such that the diagram
commutes, where the vertical map is the inclusion, and the left-most horizontal map is multiplication. The result follows.
Proposition 6.3.
Let
$Q$
denote the cokernel of the injection
$S/I_C \hookrightarrow R$
. The following hold.
-
(1) We have
$Q_{qd}=0$
for all
$q \ge 0$
. In particular, if
$0\leq j \lt d$
then any element of
$S_{d-j}$
annihilates any element of
$Q_{qd+j}$
. -
(2) For all
$e\geq 0$
, we have
$\dim Q_e = \dim Q_{e+d}$
. -
(3) The support of the sheaf
$\widetilde {Q}$
is the set of points in
$D$
. In particular,
$\widetilde {Q}$
is a zero-dimensional sheaf on
${\mathbb P}(W)$
, and
$Q$
is a one-dimensional
$S$
module.
-
(4) We have
$H^j_{\mathfrak m} Q = 0$
for
$j\ne 1$
, and
$(H^1_{\mathfrak m} Q)_e= 0$
for
$e\geq 0$
. In particular,
$Q$
is a Cohen–Macaulay
$S$
module, and its weighted regularity (Definition 4.7) is at most 0.
Before beginning the proof, we discuss a simple example.
Example 6.4. Consider Example 2.1, where
$S/I_C \cong k[s^2,st,st^3,t^4]$
and
$R \cong k[s^2,st,t^2]$
, so that
$Q=t^2 \cdot k [t^4]$
. In other words, letting
$M = S / (x_0, x_1, x_2)$
, we have
$Q \cong M(-1)$
. Observe that
$Q$
is concentrated in positive odd degrees, and each of its nonzero homogeneous components is a one-dimensional
$k$
-vector space. Its support is the point
$V(x_0, x_1, x_2)$
in
${\mathbb P}(W)$
, which is the point in
$D$
. Clearly,
$H^0_{\mathfrak m} Q=0$
because
$x_3$
is a nonzero divisor on
$Q$
. A local duality argument implies that
$H^1_{\mathfrak m} Q= t^{-2}\cdot k[t^{-4}]$
, which is zero in nonnegative degrees.
Proof of Proposition 6.3. Part (1) is clear from Lemma 6.5, and part (3) is immediate from Lemma 6.2(2). For part (2), we write
$e=qd+e'$
with
$0\leq e'\lt d$
and
$q\geq 0$
. When
$e = 0$
, this is clear from part (1). Assume
$e \gt 0$
. We have
\begin{equation*} \dim Q_e= \dim H^0(C,L^e) - \dim H^0(C,L^e \otimes {\mathcal {O}}(-e'D)) =e'\cdot \deg D, \end{equation*}
where the first equality follows from Lemmas 3.11 and 6.2(2), and the second follows from the Riemann–Roch Theorem. In particular, we see that
$\dim Q_e$
only depends on the remainder of
$e$
modulo
$d$
; this proves part (2). Finally, we consider part (4). The inclusion
$D\subseteq C$
yields a short exact sequence
\begin{equation*} 0\to {\mathcal O}_C(-D)\to {\mathcal O}_C\to {\mathcal O}_D\to 0. \end{equation*}
Twisting by
$L^d$
, and noting that
${\mathcal O}_D \otimes L^d ={\mathcal O}_D$
because
$D$
is zero dimensional, we obtain a short exact sequence
\begin{equation*} 0\to L^d \otimes {\mathcal O}_{C}(-D) \to L^d \to {\mathcal O}_D\to 0. \end{equation*}
Noting that
$H^1(C,L^d \otimes {\mathcal {O}}(-D))=0$
since
$\deg (L^d \otimes {\mathcal {O}}(-D)) \ge \deg (L \otimes {\mathcal {O}}(-D)) \ge 2g+1$
, this short exact sequence induces a surjection
\begin{equation} (S/I_C)_d \cong H^0(C,L^d) \twoheadrightarrow H^0(D,{\mathcal O}_D). \end{equation}
Since
$D$
is a finite collection of points, it is an affine scheme, and so
$H^0(D,{\mathcal O}_D)$
contains a unit. Choose a degree
$d$
element
$u\in S_d$
such that the surjection
\begin{equation*} S_d \twoheadrightarrow (S/I_C)_d \xrightarrow{(6.5)} H^0(D,{\mathcal O}_D) \end{equation*}
sends
$u$
to a unit. This implies that the map
$Q\to Q(d)$
given by multiplication by
$u$
does not alter the multiplicity of vanishing along
$D$
and, thus, induces an isomorphism
$Q_e\to Q_{e+d}$
for all
$e\geq 0$
. In particular, any nonzero element of
$Q_e$
with
$0\lt e\lt d$
cannot be annihilated by the entire maximal ideal
$\mathfrak m$
, and so
$H^0_{\mathfrak m} Q=0$
. Since
$\dim Q=1$
, we also have
$H^i_{\mathfrak m}Q = 0$
for
$i\gt 1$
. It remains to consider
$H^1_{\mathfrak m} Q$
. Using the fact that
$H^0_{\mathfrak m} Q=0$
, [Reference EisenbudEis95, Theorem A4.1] yields a short exact sequence
\begin{equation*} 0 \to Q_e \to H^0({\mathbb P}(W),\widetilde {Q(e)}) \to (H^1_{\mathfrak m} Q)_e \to 0. \end{equation*}
We know
$\dim Q_e = \dim Q_{e + d}$
for all
$e \ge 0$
. In fact, since the map
$Q(e) \xrightarrow {u} Q(e + d)$
is injective and has a finite-dimensional cokernel for all
$e \in {\mathbb {Z}}$
, we have
$\widetilde {Q(e)} \cong \widetilde {Q(e+d)}$
for all
$e \in {\mathbb {Z}}$
. It follows that
$(H^1_{\mathfrak m} Q)_e \cong (H^1_{\mathfrak m} Q)_{e+d}$
for all
$e\geq 0$
. However,
$(H^1_{\mathfrak m} Q)_e = 0$
for
$e\gg 0$
, and so we must have
$(H^1_{\mathfrak m} Q)_e = 0$
for all
$e\geq 0$
.
Proof of Theorem 1.4. From the short exact sequence
$ 0\to S/I_C \to R\to Q\to 0,$
we get a long exact sequence in local cohomology. Since
$H^0_{\mathfrak m} Q = 0$
by Proposition 6.3, and
$H^1_{\mathfrak m} R=0$
by Lemma 5.2(1), we conclude that
$H^1_{\mathfrak m} (S/I_C)=0$
. Thus,
$S/I_C$
is normally generated, and it follows from Remark 3.14(1) that
$S/I_C$
is a Cohen–Macaulay ring.
Corollary 6.6.
The weighted regularity of
$S/I_C$
and
$R$
is
$2$
if
$g\gt 0$
and
$1$
if
$g=0$
.
Proof.
By Theorem 1.4,
$S/I_C$
is a Cohen–Macaulay ring, and so
$H^0_{\mathfrak m} (S/I_C) =H^1_{\mathfrak m} (S/I_C)=0$
. Since
$R$
is a Cohen–Macaulay
$S$
module by Lemma 5.2(1), and
$Q$
is a one-dimensional
$S$
module by Proposition 6.3(3), the short exact sequence
$0\to S/I\to R\to Q\to 0$
yields the short exact sequence
$ 0\to H^1_{\mathfrak m} Q \to H^2_{\mathfrak m} (S/I_C) \to H^2_{\mathfrak m} R \to 0.$
Proposition 6.3(4) implies that
$(H^1_{\mathfrak m} Q)_e=0$
for
$e\geq 0$
, and (5.3) implies that
$(H^2_{\mathfrak m} R)_e\cong H^1(C,L^e)$
. We have
$H^1(C,L^e) = 0$
if and only if
$e\gt 0$
(respectively
$e \ge 0)$
when
$g\gt 0$
(respectively
$g=0$
). The statement immediately follows.
Proof of Theorem 1.5. Normal generation follows from Theorem 1.4. Let us record the following computation:
\begin{align*} \dim W &= \dim W_1 + \dim W_d \\ &= \dim H^0(C, L \otimes {\mathcal {O}}(-D)) + \dim H^0(C, L^d) - \dim H^0(C, L^d \otimes {\mathcal {O}}(-dD)) \\ &= \deg (L \otimes {\mathcal {O}}(-D)) - g + 1 + d\deg (D)\\ &\ge g + 2 + q + d\deg (D). \end{align*}
Here the first two equalities follows from the definition of a log complete series along with Lemma 3.11, the third from the Riemann–Roch Theorem, and the inequality by hypothesis. Also,
\begin{equation*} \dim W_1 = \dim H^0(C, L \otimes {\mathcal {O}}(-D)) = \deg (L \otimes {\mathcal {O}}(-D)) - g + 1 \ge g + 2 + q \gt g, \end{equation*}
and so the assumption
$\dim S_1 \gt g$
in Theorem 1.7 holds here.
Let
$Q$
be as in Proposition 6.3. By Theorem 1.4, Lemma 5.2(1), and Proposition 6.3(4), we have a short exact sequence
$ 0\to S/I_C \to R \to Q \to 0$
of Cohen–Macaulay
$S$
modules of dimensions
$2,2,$
and
$1$
, respectively. Recall that
$S=k[x_0, \ldots , x_n]$
and
$|{\mathbf {d}}| = \sum _{i=0}^n \deg x_i$
. Write
$\omega _{R}:= {Ext}_S^{n-1}( R, S(-|{\mathbf {d}}|))$
,
$\omega _{S/I_C}:= {Ext}_S^{n-1}( S/I_C, S(-|{\mathbf {d}}|))$
, and
$\omega _Q:={Ext}_S^{n}(Q,S(-|{\mathbf {d}}|))$
for the Matlis duals of these modules. We have a short exact sequence
\begin{equation*} 0\to \omega _R \to \omega _{S/I_C}\to \omega _Q \to 0. \end{equation*}
Just as in our proof of Theorem 1.7, we must consider the
$g=0$
and
$g \gt 0$
cases separately.
Case 1:
$g = 0$
. While we argue as in the proof of Theorem 1.7, we recapitulate the details for completeness. Since
$S/I_C$
is a Cohen–Macaulay ring of dimension 2, we have
$ \beta _{i,j}(S/I_C) = \beta _{n - 1 - i, |\mathbf {d}| - j}({\omega }_{S/I_C})$
for all
$i, j$
. Now, suppose that
$\beta _{i, j}(S/I_C) = \beta _{n-1-i, |\mathbf {d}| - j}({\omega }_{S/I_C}) \ne 0$
, and assume that
$j \gt w^{i+1}$
. We have
\begin{equation*} |\mathbf {d}| - w_{n-i} = \sum _{j=0}^n d_j - \sum _{j=0}^{n-1-i} d_j = \sum _{j=n-i}^n d_j = w^{i+1} \lt j. \end{equation*}
Rearranging, we get
$|{\mathbf {d}}| - j \lt w_{n-i}$
. Corollary 6.6 (along with local duality) implies that
$({\omega }_{S/I_C})_1 \ne 0$
and
$({\omega }_{S/I_C})_{\lt 1} = 0$
. Since
$I_C$
is prime, every variable
$x_i \in S$
is a nonzero divisor on
${\omega }_{S/I_C}$
. In particular,
$x_0$
has this property. Applying Lemma 5.1(2), with
$\ell = 0$
, we get
$|{\mathbf {d}}| -j\ge w_{n-i}$
, a contradiction. Thus, if
$\beta _{i, j}(S/I_C) \ne 0$
then
$j \le w^{i+1}$
. It follows that the embedding
$C \subseteq {\mathbb P}(W)$
satisfies the weighted
$N_p$
condition for all
$p \ge 0$
.
Case 2:
$g \gt 0$
. We first prove that
\begin{equation} \textrm {Tor}_i^S(\omega _R,k)_{j}= \textrm {Tor}_i^S(\omega _{S/I_C},k)_{j} \end{equation}
for all
$j \lt w_{i+1}$
. Proposition 6.3(4) (along with local duality) implies that
$({\omega }_{Q})_i = 0$
for
$i \le 0$
, while Corollary 6.6 (along with local duality) implies that
$({\omega }_{S/I_C})_0 \ne 0$
and
$({\omega }_{S/I_C})_{\lt 0} = 0$
, and similarly for
${\omega }_R$
. Lemma 4.6 therefore implies that the strongly linear strands of the minimal free resolutions of
${\omega }_R$
and
${\omega }_{S/I_C}$
are isomorphic. The identification (6.7) now follows by applying Lemma 5.3 to both
${\omega }_R$
and
${\omega }_{S/I_C}$
. (Note that
$\dim ({\omega }_{R})_0 = \dim ({\omega }_{S/I_C})_0 = g$
, and so, since
$\dim W_1 \gt g$
, the assumption ‘
$\dim W_1 \gt M_0$
’ in Lemma 5.3 holds for both
$M = {\omega }_R$
and
$M = {\omega }_{S/I_C}$
.) Finally, as in the proof of Theorem 1.7 (and Case 1), we have
${\beta }_{i,j}(R) = {\beta }_{n-1-i, |{\mathbf {d}}| - j}({\omega }_R)$
, and similarly for
$S/I_C$
. The equality (6.7) implies that
$\textrm {Tor}_{i}(R,k)_{j} = \textrm {Tor}_i(S/I_C,k)_{j}$
whenever
$|{\mathbf {d}}| - j \lt w_{n - i},$
i.e.
$j \gt |{\mathbf {d}}| - w_{n-i} = w^{i+1}$
. Applying Theorem 1.7, we therefore conclude that if
$i \le \dim W - g - 2$
and
${\beta }_{i,j}(S/I_C) \ne 0$
, then
$j \le w^{i+1}$
. Since
$\dim W - g - 2 \ge q + d\deg (D)$
, it follows that the embedding
$C \subseteq {\mathbb P}(W)$
satisfies the weighted
$N_{q + d\deg (D)}$
property.
7. Questions
7.1 Higher-dimensional varieties
Mumford famously showed that any high degree Veronese of a projective variety is ‘cut out by quadrics’ [Reference MumfordMum70]; see also the generalization in [Reference Sidman and SmithSS11]. Corollary 1.6 is an analogue of Mumford’s result for curves in weighted projective spaces; it is natural to ask if this result can be extended to other varieties in weighted projective spaces.
Question 7.1. Can one prove results like Corollary 1.6 for higher-dimensional varieties embedded in weighted projective spaces?
We can also ask about normal generation and the
$N_p$
conditions for higher-dimensional varieties. Here, the central results are those of [Reference Ein and LazarsfeldEL93], which prove
$N_p$
results for embeddings by line bundles of the form
$K_X+L^d+B$
, where
$K_X$
is the canonical bundle,
$L$
is very ample, and
$B$
is effective.
Question 7.2. Can one obtain
$N_p$
conditions for higher-dimensional varieties embedded by a log complete series, under hypotheses similar to those in [Reference Ein and LazarsfeldEL93]?
Embeddings into weighted spaces also provide an intermediate case for investigating asymptotic syzygy-type questions, as in [Reference Ein and LazarsfeldEL12].
Question 7.3. With notation as in Question 7.1, can one prove asymptotic nonvanishing results, similar to what happens in the main results of [Reference Ein and LazarsfeldEL12]? At the other extreme, can one prove asymptotic vanishing results as in [Reference ParkPar21]?
7.2 Scrolls and the gonality conjecture
There is a rather trivial sense in which curves embedded via log complete series of high degree satisfy an analogue of Green–Lazarsfeld’s gonality conjecture. Recall that a high degree curve in
${\mathbb P}^r$
has regularity
$2$
, and so the Betti table looks as follows.
The
$N_p$
conditions are about the moment we first get nonzero entries in row
$2$
, i.e. column
$a+1$
in the picture. In [Reference Green and LazarsfeldGL88], Green–Lazarsfeld conjectured that the moment where we first get a zero entry in row
$1$
, i.e. column
$b+1$
in the picture, is determined by the gonality
$\text {gon}(C)$
of the curve. This is the Green–Lazarsfeld gonality conjecture, and it was proven in [Reference Ein and LazarsfeldEL15], utilizing techniques originally developed by Voisin [Reference VoisinVoi02].
In the standard graded setting,
$b$
is the maximal index such that
$F_i$
has a minimal generator of degree
$i+1$
. In the weighted setting, a natural analogue of the invariant
$b$
would be to let
$b(C) := \max \{ i \text { : } F_i\ \text {has a generator of degree }w_i+1\}.$
However, the main result of [Reference Ein and LazarsfeldEL15] immediately implies that, with notation as in Theorem 1.5, we have
$b(C) = \dim W_1 - 2 -\textrm {gon}(C)$
for
$\deg L \gg 0$
. Since this only depends on the degree 1 part of
$W$
, it tells us nothing new about the relationship between the geometry of curves and the algebraic properties of syzygies. So if we want to find a meaningful weighted analogue of the gonality conjecture, we will need to look in a different direction.
The Green–Lazarsfeld gonality conjecture is one of a series of conjectures about the extent to which the syzygies of a curve
$C$
are determined by embeddings of
$C$
into special varieties such as scrolls or other varieties of minimal degree (or minimal regularity). To develop a meaningful weighted analogue of the Green–Lazarsfeld gonality conjecture, a natural first question to tackle would be as follows.
Question 7.4. Can we develop a weighted theory of rational normal scrolls, or varieties of minimal degree, or varieties of minimal regularity? More specifically, can one develop such theories for the weighted spaces
${\mathbb P}(1^a,d^b)$
that arise in Theorem 1.5?
There is a famous classical connection between varieties of minimal degree and the
$N_p$
condition: a variety has minimal degree if and only if it satisfies the
$N_p$
condition for the maximal possible
$p$
, i.e. if and only if its resolution is purely linear. If one can answer parts of Question 7.4, it would be interesting to then investigate how that answer is related to the weighted
$N_p$
conditions explored in this paper.
In a different direction, a famous result of Gruson, Lazarsfeld and Peskine [Reference Gruson, Lazarsfeld and PeskineGLP83] bounds the regularity of any nondegenerate irreducible curve
$C\subseteq {\mathbb P}^r$
in terms of its degree. It would be interesting to explore an analogue of such a theorem.
Question 7.5. Let
$C$
be a smooth (or irreducible) curve in
${\mathbb P}(d_0, \ldots , d_n)$
. Can one bound the regularity of
$I_C$
via a Gruson–Lazarsfeld–Peskine-type formula?
7.3
$M_L$
bundles
Green, Lazarsfeld and others have used positivity of
$M_L$
bundles to obtain
$N_p$
results for syzygies of a curve
$C$
embedded by a line bundle
$L$
[Reference Agostini, Küronya and LozovanuAKL19, Reference Ein and LazarsfeldEL93, Reference Green and LazarsfeldGL88, Reference Gruson, Lazarsfeld and PeskineGLP83, Reference Küronya and LozovanuKL19, Reference PareschiPar00, Reference ParkPar21]. Let
$C\subseteq {\mathbb P}^n$
be a curve embedded by the complete linear series for
$L$
. The vector bundle
$M_L$
is defined by the short exact sequence
$ 0\gets L \gets H^0(C,L)\otimes _k {\mathcal O}_C \gets M_L \gets 0.$
Vanishing results about exterior powers of
$M_L$
can be used to obtain
$N_p$
results about syzygies of the embedding
$C\hookrightarrow \mathbb P^r$
by the complete linear series
$|L|$
.
In the nonstandard graded case, the setup is more subtle, as the linear series involves sections of different degrees. This would require altering the basic framework, and it would be interesting to see whether
$N_p$
results for varieties could be proven via weighted analogues of this approach.
7.4 Stacky curves
Stacky curves have arisen in recent work on Gromov–Witten theory, mirror symmetry, the study of modular curves and more; see [Reference Voight and Zureick-BrownVZB22] and the references therein. In [Reference Voight and Zureick-BrownVZB22], Voigt and Zureick-Brown prove analogues of classical results like Petri’s Theorem for stacky curves; in fact, their results can be viewed as showing that the canonical embeddings satisfy our weighted
$N_1$
condition. Stacky curves cannot generally be embedded into standard projective space; rather, they embed into weighted projective stacks. The only relevant
$N_p$
conditions for such curves are therefore in the weighted projective setting.
Question 7.6. Prove an analogue of Theorem 1.5 for stacky curves embedded into weighted projective stacks by high degree line bundles.
We highlight one aspect where stacky curves differ from smooth curves. For a line bundle of high enough degree on a smooth curve, the rank of the global sections depends only on the degree of the line bundle. This is not the case for stacky curves, as the space of global sections also depends on the behavior of the corresponding divisor at the stacky points. So instead of simply fixing the degree of the line bundle, a more natural setup for a stacky curve might be to follow the recipe from Ein and Lazarsfeld [Reference Ein and LazarsfeldEL93] and focus on
$N_p$
conditions for divisors of the form
$K+ L^d + B$
, where
$K$
is the canonical divisor,
$L$
is very ample, and
$B$
is effective.
In a slightly different direction, one could focus on canonical embeddings. Green’s Conjecture relates classical
$N_p$
conditions to the intrinsic geometry of a canonical curve, specifically to its Clifford index. The canonical embedding of a stacky curve lands in a weighted projective stack, and thus, our weighted
$N_p$
conditions provide a natural setting for considering a stacky analogue of Green’s Conjecture.
Question 7.7. Can one use weighted
$N_p$
conditions to articulate an analogue of Green’s conjecture for stacky curves?
7.5 Nonstandard Koszul rings
Koszul rings were defined by Priddy [Reference PriddyPri70] and now play a fundamental role within commutative algebra [Reference Avramov and EisenbudAE92, Reference Avramov and PeevaAP01, Reference Avramov, Conca and IyengarACI15, Reference ConcaCon00]. One rich source of Koszul rings comes from high degree Veronese embeddings. Let
$X\subseteq {\mathbb P}^r$
be a smooth variety embedded by a complete linear series for
$L^d$
, where
$L$
is very ample and
$d\gg 0$
; it is known that the homogeneous coordinate ring of
$X\subseteq {\mathbb P}^r$
is a Koszul ring [Reference BackelinBac86, Reference Eisenbud, Reeves and TotaroERT94].
It would be interesting to know if high degree embeddings into weighted spaces (via a log complete series) can provide more exotic examples of Koszul rings, or related concepts. The following example shows some of the subtle behavior that might arise.
Example 7.8. Let
${\mathbb P}^1 \to {\mathbb P}(1^3,2^2)$
be the map
$[s:t]\mapsto [s^3:s^2t:st^2:st^5:t^6]$
be the map determined by the log complete series for
${\mathcal O}_{{\mathbb P}^1}(3)$
with
$d=2$
and
$\deg (D)=1$
. Let
$S=k[x_0,x_1,x_2,y_0,y_1]$
be the Cox ring of
${\mathbb P}(1^3,2^2)$
. The defining ideal of the image is
$ I = \langle x_{1}^{2}-x_{0}x_{2},\,x_{2}y_{0}-x_{1}y_{1},\,x_{ 1}y_{0}-x_{0}y_{1},\,x_{2}^{3}-x_{0}y_{1},\,x_{1}x_{2}^{2}-x_{0}y_{0},\,y _{0}^{2}-x_{2}^{2}y_{1} \rangle .$
The ring
$S/I_C$
is isomorphic to the subalgebra
$k[s^3,s^2t,st^2,st^5,t^6]$
, and it is a variant of a Veronese subring; for instance, it contains the degree
$6$
Veronese subring. The ring
$T = S/I_C$
does not satisfy the standard definition of a graded Koszul ring, as the minimal free resolution of the residue field has the form
$[T \gets T(-1)^3 \oplus T(-2)^2 \leftarrow \cdots ]$
. However, if we consider the grevlex order with
$y_0\gt y_1\gt x_0\gt x_1\gt x_2$
then the initial ideal is
$ \textrm {in}(I) = \langle x_{1}^{2},\,x_{1}y_{1},\,x_1y_{0},\,x_0y_{1},\,y_{0}x_{0},\,y_{0}^{2}\rangle .$
Thus,
$I$
has a quadratic Gröbner basis; if
$S$
were standard graded then this would imply that
$S/I_C$
is
$G$
quadratic and, therefore, Koszul [Reference ConcaCon00]. Given the nonstandard grading, it implies that
$S/I_C$
is a sort of nonstandard graded deformation of a Koszul ring.
Question 7.9. Let
$X$
be a smooth variety, and consider an embedding
$X\hookrightarrow {\mathbb P}(W)$
given by a log complete series for
$L^e$
, where
$L$
is very ample and
$e \gg 0$
. Let
$I_X\subseteq S$
be the defining ideal in the corresponding nonstandard graded polynomial ring. Does
$I_X$
admit a quadratic Gröbner basis? What sort of Koszul-type properties are satisfied by
$S/I_X$
?
7.6
$N_p$
conditions for curves in other toric varieties
Another natural direction is to ask whether smooth curves in other toric varieties also satisfy
$N_p$
conditions. To approach this, one must consider the following.
Question 7.10. Let
$S$
be the
${\mathbb Z}^r$
-graded Cox ring of a simplicial toric variety
$X$
and
$B$
the corresponding irrelevant ideal.
-
(1) What is a good analogue of a complete, or log complete, linear series?
-
(2) What is a good analogue of normal generation?
-
(3) What is the appropriate analogue of the
$N_p$
conditions in this setting? -
(4) Does the answer to (1) or (2) depend only on the grading of
$S$
, or does it also depend on the choice of the irrelevant ideal
$B$
? -
(5) When defining the
$N_p$
conditions, should one focus on minimal free resolutions or on virtual resolutions?
Even for a product of projective spaces, some of these questions are open.
Question 7.11. Can one develop analogues of the main results of this paper for a smooth curve in a product of projective spaces?
Acknowledgements
We thank David Eisenbud, Jordan Ellenberg, Matthew Satriano, Hal Schenck, Frank-Olaf Schreyer, and Greg Smith for valuable conversations. We also thank the referee for many helpful comments.
Conflicts of interest
None.
Financial support
MKB was supported by NSF-RTG grant 1502553 and DE by NSF grant DMS-2200469.
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