2.1. q-bic forms and hypersurfaces

A q-bic form on V is a nonzero linear map

$$\begin{align*}\beta \colon V^{[1]} \otimes_{\mathbf{k}} V \to \mathbf{k}, \end{align*}$$

where here and elsewhere, is the q-power Frobenius twist of V. Let $\mathbf {P} V$ be the projective space of lines associated with V. The q-bic equation associated with $\beta $ is the map

obtained by pairing the canonical section $\mathrm {eu} \colon \mathcal {O}_{\mathbf {P} V}(-1) \to V \otimes _{\mathbf {k}} \mathcal {O}_{\mathbf {P} V}$ with its q-power via $\beta $ . The q-bic hypersurface associated with the q-bic form $\beta $ is the hypersurface defined by $f_\beta $ and may be identified as the space of lines in V isotropic for $\beta $ :

A choice of basis $V = \langle e_0, \ldots , e_n \rangle $ makes this more explicit: Writing for the corresponding projective coordinates on $\mathbf {P} V \cong \mathbf {P}^n$ , and for the $(i,j)$ th entry of the Gram matrix of $\beta $ with respect to the given basis, as in [Reference Cheng9, Section 1.2], the q-bic equation $f_\beta $ is the homogeneous polynomial of degree $q+1$ given by

From any of the descriptions, it is easy to check that a hyperplane section of a q-bic hypersurface is a q-bic hypersurface (see [Reference Cheng10, Lemma 1.9]).

To illustrate the utility of distinguishing the shape of the q-bic equation , and to record the computation for later use, the following shows that the action of the q-power absolute Frobenius morphism $\operatorname {\mathrm {Fr}} \colon X \to X$ is zero on higher cohomology.

Proof. Consider the commutative diagram of abelian sheaves on $\mathbf {P} V$ given by

where $\operatorname {\mathrm {Fr}}$ acts on local sections by $s \mapsto s^q$ . Taking cohomology yields a commutative diagram

Compute the map $f^{q-1} \operatorname {\mathrm {Fr}}$ on the right as follows: Upon choosing coordinates $(x_0:\cdots :x_n)$ for $\mathbf {P} V = \mathbf {P}^n$ , a basis for $\mathrm {H}^n(\mathbf {P} V, \mathcal {O}_{\mathbf {P} V}(-q-1))$ is given by elements

Monomials appearing in $f^{q-1}$ are of the form $a^q b$ , where a and b are themselves monomials of degree $q-1$ . Write $a = x_0^{a_0} \dots x_n^{a_n}$ with $a_0 + \cdots + a_n = q-1$ . Then $f^{q-1}\operatorname {\mathrm {Fr}}(\xi )$ is a sum of terms

$$\begin{align*}a^q b \cdot (x_0^{i_0} \dots x_n^{i_n})^{-q} = b \cdot (x_0^{i_0 - a_0} \dots x_n^{i_n - a_n})^{-q}. \end{align*}$$

Since $(i_0 - a_0) + \cdots + (i_n - a_n) = 2 \leq n$ , there is some index j such that $i_j - a_j \leq 0$ , and so this is $0$ . Each potential contribution to $f^{q-1}\operatorname {\mathrm {Fr}}(\xi )$ vanishes, and so $f^{q-1}\operatorname {\mathrm {Fr}}(\xi ) = 0$ .

2.3. Classification

Isomorphic q-bic forms yield projectively equivalent q-bic hypersurfaces. Over an algebraically closed field $\mathbf {k}$ , [Reference Cheng9, Theorem A] shows that there are finitely many isomorphism classes of q-bic forms on V, and that they may be classified via their Gram matrices: there exists a basis $V = \langle e_0,\ldots ,e_n \rangle $ and integers $a, b_m \in \mathbf {Z}_{\geq 0}$ such that

$$\begin{align*}\operatorname{\mathrm{Gram}}(\beta;e_0,\ldots,e_n) = \mathbf{1}^{\oplus a} \oplus \Big(\bigoplus\nolimits_{m \geq 1} \mathbf{N}_m^{\oplus b_m}\Big), \end{align*}$$

where $\mathbf {1}$ is the $1$ -by- $1$ matrix with unique entry $1$ , $\mathbf {N}_m$ is the m-by-m Jordan block with $0$ ’s on the diagonal, and $\oplus $ denotes block diagonal sums of matrices. The tuple $(a; b_m)_{m \geq 1}$ is called the type of $\beta $ , and is the fundamental invariant of the form; the sum $\sum _{m \geq 1} b_m$ is its corank. When q-bic forms are varied in a family, the corank can jump and the type can change; the Hasse diagram of specialization relations is not linear in general, and some partial relations are given in [Reference Cheng9, Theorem C].

Of note is the $\mathbf {N}_1^{\oplus b_1}$ piece, which is a $b_1$ -by- $b_1$ zero matrix, and will sometimes denoted by $\mathbf {0}^{\oplus b_1}$ . Its underlying subspace is the radical of $\beta $ :

It is straightforward to check that X is a cone if and only if $b_1 \neq 0$ , and that the vertex is the projectivization of the radical (see also [Reference Cheng8, Section 2.4] for an invariant treatment).

2.4. q-bic points and curves

In low-dimensions, the classification and specialization relations are quite simple and, especially when combined with the fact that hyperplane sections of q-bics are q-bics, are geometrically very useful (see [Reference Cheng8, Chapter 3] for more). When V has dimension $2$ , the possible types of q-bic forms and their specialization relations are

$$\begin{align*}\mathbf{1}^{\oplus 2} \rightsquigarrow \mathbf{N}_2 \rightsquigarrow \mathbf{0} \oplus \mathbf{1}, \end{align*}$$

corresponding to the subschemes in $\mathbf {P}^1$ defined by $x_0^{q+1} + x_1^{q+1}$ , $x_0^q x_1$ , and $x_1^{q+1}$ . Thus, q-bic points are either $q+1$ reduced points, a reduced point with a q-fold point, or one point of multiplicity $q+1$ .

When V is three-dimensional, the types and specializations excluding cones over points are

$$\begin{align*}\mathbf{1}^{\oplus 3} \rightsquigarrow \mathbf{1} \oplus \mathbf{N}_2 \rightsquigarrow \mathbf{N}_3. \end{align*}$$

The corresponding subschemes of $\mathbf {P}^2$ may be described as: a smooth curve of degree $q+1$ ; an irreducible, geometrically rational curve with a single unibranch singularity; and a reducible curve with a linear component meeting an irreducible, geometrically rational component of degree q at its unique unibranch singularity.

2.5. Smoothness

A q-bic hypersurface X is smooth if and only if its underlying q-bic form $\beta $ is nonsingular in the sense that one—equivalently, both—of the maps $\beta \colon V \to V^{[1],\vee }$ or $\beta ^\vee \colon V^{[1]} \to V^\vee $ is an isomorphism; with a choice of basis, this is equivalent to invertibility of a Gram matrix. The singular locus of X is supported on the projectivization of the subspace

the Frobenius descent of the left kernel (see [Reference Cheng10, Corollary 2.6]). The embedded tangent space $\mathbf {T}_{X,x} \subset \mathbf {P} V$ to X at a point $x = \mathbf {P} L$ is the projectivization of

the right orthogonal of L (see [Reference Cheng10, Proposition 2.2]).

2.6. Hermitian structures

A vector $v \in V$ is said to be Hermitian if

$$\begin{align*}\beta(u^{[1]}, v) = \beta(v^{[1]}, u)^q \;\;\text{for all}\; u \in V. \end{align*}$$

A subspace $U \subset V$ is Hermitian if it is spanned by Hermitian vectors. The Hermitian equation implies that the left and right orthogonals of U,

coincide. When $\beta $ is nonsingular, there are only finitely many Hermitian vectors, whence finitely many Hermitian subspaces, and they span V (see [Reference Cheng9, Sections 2.1–2.6]).

Continuing with $\beta $ nonsingular, there is a canonical $\mathbf {k}$ -vector space isomorphism

between the $q^2$ -Frobenius twist of V and V itself; this is an $\mathbf {F}_{q^2}$ descent datum on V. Pre-composing with the universal $q^2$ -linear map $V \to V^{[2]}$ yields a $q^2$ -linear endomorphism $\phi \colon V \to V$ . It preserves isotropicity so it induces an endomorphism $\phi _X \colon X \to X$ . In a basis $V = \langle e_0, \ldots , e_n \rangle $ of Hermitian vectors, the Gram matrix of $\beta $ is Hermitian in the sense that

$$\begin{align*}\operatorname{\mathrm{Gram}}(\beta; e_0,\ldots,e_n)^\vee = \operatorname{\mathrm{Gram}}(\beta; e_0,\ldots,e_n)^{[1]}, \end{align*}$$

from which it follows that $\sigma _\beta $ is represented by the identity matrix, and $\phi _X$ is the $q^2$ -power geometric Frobenius morphism in the corresponding coordinates (see [Reference Cheng10, Section 4.3]).

By [Reference Cheng10, Lemma 1.3], the fixed set of $\phi _X$ is the subset $X_{\mathrm {Herm}}$ of Hermitian points of X, consisting of those points $x = \mathbf {P} L,$ where L is spanned by a Hermitian vector. Comparing with the construction of $\phi _X$ , this implies that the set of Hermitian points of X is the zero locus of the map

$$\begin{align*}\tilde\theta \colon \mathcal{O}_X(-q^2) \stackrel{\mathrm{eu}^{[2]}}{\longrightarrow} V^{[2]} \otimes_{\mathbf{k}} \mathcal{O}_X \stackrel{\sigma_\beta}{\longrightarrow} V \otimes_{\mathbf{k}} \mathcal{O}_X \longrightarrow \mathcal{T}_{\mathbf{P} V}(-1)\rvert_X, \end{align*}$$

where the final arrow arises from the dual Euler sequence. The following shows that this map factors through the tangent bundle of X. A geometric description of $\phi _X(x)$ may be extracted from the proof below, as can be found in [Reference Cheng8, Lemma 2.9.9], and this is most clear when X is a curve: the tangent line $\mathbf {T}_{X,x}$ intersects X at x with multiplicity q, and the residual point of intersection is $\phi _X(x)$ .

Proof. The tangent bundle of X is the kernel of the normal map $\delta \colon \mathcal {T}_{\mathbf {P} V}(-1)\rvert _X \to \mathcal {N}_{X/\mathbf {P} V}(-1)$ , induced via the dual Euler sequence by the map $\tilde \delta \colon V \otimes _{\mathbf {k}} \mathcal {O}_X \to \mathcal {N}_{X/\mathbf {P} V}(-1)$ which takes directional derivatives of a fixed equation of X. A simple computation shows that there is a commutative diagram

and so $\delta \circ \tilde \theta $ factors through the map

$$\begin{align*}\tilde\delta \circ \sigma_\beta \circ \mathrm{eu}^{[2]} = \mathrm{eu}^{[1],\vee} \circ \beta^{[1],\vee} \circ \mathrm{eu}^{[2]} = \beta(\mathrm{eu}^{[1]}, \mathrm{eu})^q = 0, \end{align*}$$

see [Reference Cheng9, Section 1.7] for the penultimate relation. Thus, $\tilde {\theta }$ factors through $\mathcal {T}_X(-1)$ .

2.8. Scheme of lines

The Fano scheme of lines of a q-bic hypersurface X may be viewed as the space of two-dimensional subspaces in V which are totally isotropic for the q-bic form $\beta $ :

This description exhibits $\mathbf {F}$ as the zero locus in the Grassmannaian of

$$\begin{align*}\beta_{\mathcal{S}} \colon \mathcal{S}^{[1]} \otimes_{\mathcal{O}_{\mathbf{G}}} \mathcal{S} \to \mathcal{O}_{\mathbf{G}} \end{align*}$$

induced by restricting $\beta $ to the universal rank $2$ subbundle $\mathcal {S} \subset V \otimes _{\mathbf {k}} \mathcal {O}_{\mathbf {G}}$ . A direct construction shows that $\mathbf {F}$ is nonempty whenever $n \geq 3$ , so this implies that $\dim \mathbf {F} \geq 2n-6$ (see [Reference Cheng10, Sections 1.10–1.12]). Furthermore, $\mathbf {F}$ is connected whenever $n \geq 4$ (see [Reference Cheng10, Proposition 2.9]).

When $\mathbf {F}$ is generically smooth and of expected dimension $2n-6$ , it is a local complete intersection scheme whose structure sheaf admits a Koszul resolution by $\wedge ^* \mathcal {S}^{[1]} \otimes _{\mathcal {O}_{\mathbf {G}}} \mathcal {S}$ , and its dualizing sheaf may be computed to be

$$\begin{align*}\omega_{\mathbf{F}} \cong \mathcal{O}_{\mathbf{F}}(2q+1-n) \otimes_{\mathbf{k}} \det(V)^{\vee, \otimes 2}, \end{align*}$$

where the twist by $\det (V)$ is useful for tracking weights of algebraic group actions (see [Reference Cheng10, Corollary 2.4]). Here, $\mathcal {O}_{\mathbf {F}}(1)$ is the Plücker line bundle; its degree is computed in [Reference Cheng10, Proposition 1.15] as

$$\begin{align*}\deg\mathcal{O}_{\mathbf{F}}(1) = \frac{(2n-6)!}{(n-1)!(n-3)!} (q+1)^2\big((n-1)q^2 + (2n-8)q + (n-1)\big). \end{align*}$$

The Fano scheme $\mathbf {F}$ is singular along the subscheme of lines through the singular locus of X:

In particular, $\mathbf {F}$ is smooth if and only if X itself is smooth, or equivalently, when $\beta $ is nonsingular (see [Reference Cheng10, Corollary 2.7]). This description moreover implies that $\mathbf {F}$ is of expected dimension if and only if the locus of lines through the singular locus of X has dimension at most $2n-6$ .

2.9. q-bic surfaces

To complete the tour in low dimensions, consider a q-bic hypersurface X of dimension $2$ , or briefly, a q-bic surface. Classification and specialization of types look like

where, notably, the Hasse diagram of specialization relations is not linear (see [Reference Cheng8, Section 3.8] for more).

The general statements of 2.8 show that X always contains lines, and that its Fano scheme $\mathbf {F}$ has degree $(q+1)(q^3+1)$ when it is of expected dimension $0$ . This means, in particular, that a smooth q-bic surface has exactly $(q+1)(q^3+1)$ distinct lines. The configuration of these lines is fascinating and is well-studied (see, e.g., [Reference Brosowsky, Page, Ryan and Smith7], [Reference Hirschfeld23], [Reference Schütt, Shioda and van Luijk45]). Other aspects of the geometry of smooth q-bic surfaces may be found, for example, in [Reference Ojiro41], [Reference Shioda49].

2.10. q-bic threefolds

In the remainder of the article, X denotes a three-dimensional q-bic hypersurface, or a q-bic threefold for short. The classification and specialization relations of q-bic threefolds, at least up to the first few cones, look as follows:

Notice that all non-cones have corank at most $2$ , which by the discussion of 2.5, implies that the singular locus of a q-bic threefold which is not a cone has dimension at most $1$ .

The Fano scheme of lines of a q-bic threefold has expected dimension $2$ by 2.8. It is easy to check that when X has corank at most $1$ and is not a cone, the singular locus of S has dimension at most $1$ . Thus, in all such cases, and also in the case X is smooth, S is generically smooth of expected dimension $2$ (see also [Reference Cheng10, Lemma 2.8]). In any of these cases, but especially when X is either smooth or of type $\mathbf {1}^{\oplus 3} \oplus \mathbf {N}_2$ , S will be referred to as the Fano surface of X.

Begin by considering the Fano surface S of a smooth q-bic threefold X, describing the tangent bundle $\mathcal {T}_S$ , and computing its Chern numbers.

Lemma 2.11. If X is smooth, then $\mathcal {T}_S \cong \mathcal {S} \otimes _{\mathcal {O}_S} \mathcal {O}_S(2-q)$ and the Chern numbers of S are

$$\begin{align*}c_1(S)^2 = (q+1)^2(q^2+1)(2q-3)^2 \;\;\text{and}\;\; c_2(S) = (q+1)^2(q^4 - 3q^3 + 4q^2 - 4q + 3). \end{align*}$$

Proof. Identify the tangent bundle of S as in [Reference Cheng10, Proposition 2.2] as $\mathcal {T}_S \cong {\mathcal {H}\!\mathit {om}}_{\mathcal {O}_S}(\mathcal {S},\mathcal {S}^{[1],\perp }/\mathcal {S})$ , where the sheaf in the target of the ${\mathcal {H}\!\mathit {om}}$ is characterized by the short exact sequence

$$\begin{align*}0 \to \mathcal{S}^{[1],\perp}/\mathcal{S} \to \mathcal{Q} \xrightarrow{\beta} \mathcal{S}^{[1],\vee} \to 0, \end{align*}$$

where $\mathcal {Q}$ is the tautological quotient sheaf. Taking determinants shows $\mathcal {S}^{[1],\perp }/\mathcal {S} \cong \mathcal {O}_S(1-q)$ , and combined with the wedge product isomorphism $\mathcal {S}^\vee \cong \mathcal {S} \otimes _{\mathcal {O}_S} \mathcal {O}_S(1)$ gives the identification of $\mathcal {T}_S$ . To compute the first Chern number, note that the Plücker line bundle $\mathcal {O}_S(1)$ has degree $(q+1)^2(q^2+1)$ by taking $n = 4$ in the formula given in 2.8. Combined with the identification of $\mathcal {T}_S$ , this gives

For the second Chern number, observe that

$$ \begin{align*} c_2(\mathcal{T}_S) & = c_2(\mathcal{S}^\vee) + c_1(\mathcal{S}^\vee) c_1(\mathcal{O}_S(1-q)) + c_1(\mathcal{O}_S(1-q))^2 \\ & = c_2(\mathcal{S}^\vee) + (q-2)(q-1) c_1(\mathcal{O}_S(1))^2. \end{align*} $$

Since a general section of $\mathcal {S}^\vee $ cuts out the scheme of lines contained in a general hyperplane section of X, which is but a smooth q-bic surface by 2.1 and 2.3, the degree of $c_2(\mathcal {S}^\vee )$ is $(q+1)(q^3+1)$ by 2.9. Combining with the degree computation for $\mathcal {O}_S(1)$ gives $c_2(S)$ .

This computation implies that the Fano surface S cannot lift to either the second Witt vectors of $\mathbf {k}$ or any characteristic $0$ base as soon as $q> 2$ . By contrast, observe that S does lift when $q = 2$ , as is seen by taking any lift of X and by applying the Fano scheme construction in families.

Proof. Dualizing the tangent bundle computation of 2.11 shows $\Omega ^1_S \cong \mathcal {S}^\vee \otimes _{\mathcal {O}_S} \mathcal {O}_S(q-2)$ . Since $\mathcal {S}^\vee $ has sections, this implies that Kodaira–Akizuki–Nakano vanishing fails on S, so [Reference Deligne and Illusie13, Corollaire 2.8] shows that S cannot lift to $\mathrm {W}_2(\mathbf {k})$ . The Chern number computation of 2.11 gives

$$\begin{align*}c_1(S)^2 - 3c_2(S) = q^2(q+1)^2(q^2-3q+1). \end{align*}$$

The quadratic formula shows that this is positive whenever $q> 2$ , and so the Bogomolov–Miyaoka–Yau inequality of [Reference Miyaoka38, Theorem 4] is not satisfied on S. Since Chern numbers are constant in flat families, this implies that S cannot lift to characteristic $0$ .

The Chern number computation also gives a simple way to compute the Euler characteristic of the structure sheaf $\mathcal {O}_S$ whenever S is smooth; this holds more generally, whenever S has expected dimension, by comparing with the Koszul resolution of $\mathcal {O}_S$ on $\mathbf {G}$ .

Proposition 2.13. If S is of its expected dimension $2$ , then

$$\begin{align*}\chi(S,\mathcal{O}_S) = \frac{1}{12}(q+1)^2(5q^4 - 15q^3 + 17q^2 - 16q + 12). \end{align*}$$

Proof. When X is a smooth, so is S, and so Noether’s formula, as in [Reference Fulton16, Example 15.2.2], applies to give the first equality in

$$\begin{align*}\chi(S,\mathcal{O}_S) = \frac{1}{12} (c_1(S)^2 + c_2(S)) = \sum\nolimits_{i = 0}^4 (-1)^i \chi(\mathbf{G}, \wedge^i \mathcal{S}^{[1]} \otimes_{\mathcal{O}_{\mathbf{G}}} \mathcal{S}). \end{align*}$$

Substituting the Chern number computations from 2.11 gives the formula in the statement. The second equality arises from taking the Koszul resolution of $\mathcal {O}_S$ on $\mathbf {G}$ . Since the Koszul resolution persists whenever S is of dimension $2$ , this gives the general case.

3.1. Cone situation

A triple $(X,\infty ,\mathbf {P}\breve {W})$ consisting of a q-bic threefold X, a point $\infty = \mathbf {P} L$ of X, and a hyperplane $\mathbf {P}\breve {W}$ such that $X \cap \mathbf {P}\breve {W}$ is a cone with vertex $\infty $ over a reduced q-bic curve $C \subset \mathbf {P}\overline {W}$ is called a cone situation; here and later, and . This is sometimes considered with some of the following additional geometric assumptions:

1. (i) there does not exist a $2$ -plane contained in X which passes through $\infty $ ; or

2. (ii) $\infty $ is a smooth point of X; or

3. (iii) C is a smooth q-bic curve.

Conditions (ii) and (iii) together imply (i). That $\infty $ is a vertex for $X \cap \mathbf {P}\breve {W}$ means that L lies in the radical of $\beta \rvert _{\breve {W}}$ , so that $\breve {W}$ is contained in both orthogonals of L (see 2.3). In particular, this means that $\mathbf {P}\breve {W}$ is contained in the embedded tangent space $\mathbf {T}_{X,\infty } = \mathbf {P} L^{[1],\perp }$ of X at $\infty $ (see 2.5); and if (ii) is satisfied, then equality $\mathbf {P}\breve {W} = \mathbf {T}_{X,\infty }$ necessarily holds.

3.2. Examples

The following are some examples of cone situations:

1. (i) Let X be a smooth q-bic threefold and let $\infty \in X$ be a Hermitian point. Then $(X,\infty ,\mathbf {T}_{X,\infty })$ is a cone situation, and it follows from 2.3 and 2.6 that all cone situations for smooth X arise this way. All the conditions (i)–(iii) are satisfied.

The next three examples pertain to q-bic threefolds of type $\mathbf {1}^{\oplus 3} \oplus \mathbf {N}_2$ . For concreteness, let

$$\begin{align*}X = \mathrm{V}(x_0^q x_1 + x_0 x_1^q + x_2^{q+1} + x_3^q x_4) \subset \mathbf{P}^4, \end{align*}$$

and set in the $\mathbf {P}^2$ in which $x_3$ and $x_4$ are projected out. Write , , and .

1. (ii) $(X,x_-,\mathbf {T}_{X,x_-})$ is a cone situation over C satisfying each of the conditions (i–(iii).

2. (iii) $(X,x_+,\mathrm {V}(x_3))$ is a cone situation over the C satisfying (i) and (iii), but not (ii).

3. (iv) $(X,\infty ,\mathbf {T}_{X,\infty })$ is a cone situation over $\mathrm {V}(x_2^{q+1} + x_3^q x_4)$ satisfying (i) and (ii), but not (iii).

The remaining examples illustrate the range of possibilities for the cone situation.

1. (v) Let and . Then $(X,\infty ,\mathbf {T}_{X,\infty })$ is a cone situation over $\mathrm {V}(x_0^q x_1 + x_1^q x_2)$ satisfying (ii), but not (i) nor (iii).

2. (vi) Let X be of type $\mathbf {0} \oplus \mathbf {1}^{\oplus 4}$ , $\infty $ the vertex of X, and $\mathbf {P}\breve {W}$ a general hyperplane through $\infty $ . Then $(X,\infty ,\mathbf {P}\breve {W})$ is a cone situation satisfying (iii), but not (i) nor (ii).

3. (vii) Let X be of type $\mathbf {0}^{\oplus 2} \oplus \mathbf {1} \oplus \mathbf {N}_2$ , $\infty $ any point of the vertex of X, and $\mathbf {P}\breve {W}$ be any hyperplane intersecting the vertex of X exactly at $\infty $ . Then $(X,\infty ,\mathbf {P}\breve {W})$ is a cone situation satisfying none of (i), (ii), nor (iii).

The following illustrates how the hypotheses of 3.1 translate to geometric consequences on X and its Fano scheme S of lines; the first statement gives sufficient conditions—although not necessary, as shown by 3.2 (ii)—for when S is of expected dimension $2$ , whereas the second classifies cone situations satisfying the two smoothness hypotheses on $(X,\infty ,\mathbf {P}\breve {W})$ .

Proof. Suppose that $(X, \infty , \mathbf {P}\breve {W})$ satisfies 3.1(i) and 3.1(ii). Then X is not a cone: otherwise, its vertex x, which is different from the smooth point $\infty $ , would span a plane $\langle x, \ell \rangle $ through $\infty $ with any line $\ell \subset X \cap \mathbf {P}\breve {W}$ through $\infty $ and not passing through x. The comments in 2.10 show that $\dim \operatorname {\mathrm {Sing}} X \leq 1$ . Suppose $x \in X$ is a singular point with underlying linear space $K \subset V$ . Lines in X through x are contained in

$$\begin{align*}X \cap \mathbf{P} K^{[1],\perp} \cap \mathbf{P} K^{\perp,[-1]} = X \cap \mathbf{T}_{X,x} \cap \mathbf{P} K^{\perp,[-1]} = X \cap \mathbf{P} K^{\perp, [-1]} \end{align*}$$

since the underlying linear space pairs to $0$ with K on either side of $\beta $ (see also [Reference Cheng10, Lemma 3.1]). Since x is not a vertex, this is a surface which is a cone with vertex x over a q-bic curve. Thus, the locus of lines through x is of dimension $1$ . Therefore, discussion of 2.8 implies that $\dim S = 2$ .

Suppose now that $(X,\infty , \mathbf {P}\breve {W})$ furthermore satisfies 3.1(iii). If X itself is smooth, then it must be 3.2(i), so suppose X is singular. Since $\infty $ is a smooth point, and since $\mathbf {P}\breve {W} = \mathbf {T}_{X,\infty }$ intersects X at a cone with a smooth base, the singular locus of X must be a single point $x = \mathbf {P} K$ disjoint from $\mathbf {P}\breve {W}$ . But $W = L^{[1],\perp }$ , so this means that the natural map $K \to L^{[1],\vee }$ is nonzero, whence an isomorphism. Splitting $V = \breve {W} \oplus K$ , it now follows that

$$\begin{align*}V^{[1],\perp} = \ker(\breve{W} \oplus K \to \breve{W}^{[1],\vee}) = \ker(\breve{W} \to \overline{W}^{[1],\vee}) = L. \end{align*}$$

Thus, the restriction of $\beta $ to is of type $\mathbf {N}_2$ . Since $U^{[1],\perp } = L^{[1],\perp }$ and $U^\perp = K^\perp $ are distinct hyperplanes, their intersection in V gives the orthogonal complement to U. Whence X is of type $\mathbf {1}^{\oplus 3} \oplus \mathbf {N}_2$ and $\infty $ is the smooth point of the subform $\mathbf {N}_2$ .

Let $(X,\infty ,\mathbf {P}\breve {W})$ be a cone situation over the q-bic curve C and let S be the Fano scheme of lines of X. Then there is a canonical closed immersion $C \hookrightarrow S$ identifying it with the closed subscheme

of lines containing $\infty $ and contained in $X \cap \mathbf {P}\breve {W}$ , so that the Plücker polarization $\mathcal {O}_S(1)$ pulls back to the planar polarization $\mathcal {O}_C(1)$ . The hypotheses of 3.1 affect $C_{\infty ,\mathbf {P}\breve {W}}$ as follows.

Proof. Establish (i) through its contrapositive: To see “ $\supseteq $ ,” if there were a line $\ell \subset X \cap \mathbf {P}\breve {W}$ not containing $\infty $ , then the cone $X \cap \mathbf {P}\breve {W}$ would contain the plane $\langle \infty , \ell \rangle $ spanned by $\infty $ and $\ell $ . To see “ $\subseteq $ ,” if there were a line $\ell \ni \infty $ not contained in $\mathbf {P}\breve {W}$ , then the orthogonals of L contain not only $\breve {W}$ , by the comments of 3.1, but also the subspace underlying $\ell $ . Therefore, L lies in the radical of $\beta $ , so X is a cone over a q-bic surface with vertex containing $\infty $ . The result now follows since every q-bic surface contains a line by 2.9.

For (ii), the comments following 3.1 imply that $\mathbf {P}\breve {W}$ is the embedded tangent space of X at $\infty $ . Since any line in X through $\infty $ is necessarily contained in $\mathbf {T}_{X,\infty }$ , the moduli problem underlying $C_\infty $ is a closed subfunctor of that of $C_{\infty , \mathbf {P}\breve {W}}$ . Since the reverse inclusion is clear, equality holds.

For (iii), observe that there is a short exact sequence of locally free sheaves

$$\begin{align*}0 \to \mathcal{T}_C \to \mathcal{T}_S\rvert_C \to \mathcal{N}_{C/S} \to 0, \end{align*}$$

where $C_{\infty ,\mathbf {P}\breve {W}}$ is identified with C: indeed, this is because the assumptions together with the discussion in 2.8 imply that S is smooth along C. Taking determinants then gives the normal bundle.

The cone situation gives a way of transforming lines in X to points of C, inducing a rational map $S \dashrightarrow C$ in good cases. Some hypothesis on $(X,\infty ,\mathbf {P}\breve {W})$ is necessary to exclude, for instance, example 3.2 (iv), wherein the locus of lines through $\infty $ forms an irreducible component of S. In the following, write $\operatorname {\mathrm {proj}}_\infty \colon \mathbf {P} V \dashrightarrow \mathbf {P}\overline {V}$ for the linear projection away from $\infty $ , and let .

Lemma 3.5. If the cone situation $(X,\infty ,\mathbf {P}\breve {W})$ satisfies 3.1(i), then there is a rational map $\varphi \colon S \dashrightarrow C$ given on points $[\ell ] \in S^\circ $ by

$$\begin{align*}\varphi([\ell]) = \operatorname{\mathrm{proj}}_\infty(\ell \cap \mathbf{P}\breve{W}) = \operatorname{\mathrm{proj}}_\infty(\ell) \cap \mathbf{P}\overline{W}. \end{align*}$$

3.6. Projection and intersection

Toward a resolution of $\varphi $ , consider the scheme parameterizing triples

consisting of a point $y \in \mathbf {P}\breve {W}$ , a line $\ell _0 \subset \mathbf {P}\overline {V}$ , and a point $y_0 \in \ell _0 \cap C$ , such that y is contained in the line in $\mathbf {P}\breve {W}$ determined by $y_0$ . Equivalently, $\mathbf {P}$ is the product $\mathbf {P} \mathcal {V}_1 \times _C \mathbf {P} \mathcal {V}_2$ of projective bundles over C, where $\mathcal {V}_1$ and $\mathcal {V}_2$ are defined via pullback of the left square and pushout of the right square, respectively, of the following commutative diagram:

where , , and similarly for the others. In particular, there are split short exact sequences

$$\begin{align*}0 \to L_C \to \mathcal{V}_1 \to \mathcal{O}_C(-1) \to 0 \quad\text{and}\quad 0 \to \mathcal{T} \to \mathcal{V}_2 \to (V/\breve{W})_C \to 0. \end{align*}$$

Let $\mathbf {P}^\circ $ be the open subscheme of $\mathbf {P}$ , where $y \neq \infty $ and $\ell _0 \not \subset \mathbf {P}\overline {W}$ . Its closed complement is the union of two effective Cartier divisors, with points

An equation of is as follows: Let $\pi _i \colon \mathbf {P}\mathcal {V}_i \to C$ and $\pi \colon \mathbf {P} \to C$ be the structure morphisms, let $\operatorname {\mathrm {pr}}_i \colon \mathbf {P} \to \mathbf {P} \mathcal {V}_i$ be the projections, and for any $\mathcal {O}_{\mathbf {P}}$ -module $\mathcal {F}$ and $a,b \in \mathbf {Z}$ , write

Then , where $u_i$ is the map obtained by composing the Euler section $\mathrm {eu}_{\pi _i}$ with the quotient map for $\mathcal {V}_i$ in the two sequences above:

$$\begin{align*}u_1 \colon \mathcal{O}_{\mathbf{P}}(-1,0) \hookrightarrow \pi^*\mathcal{V}_1 \twoheadrightarrow \pi^*\mathcal{O}_C(-1) \quad\text{and}\quad u_2 \colon \mathcal{O}_{\mathbf{P}}(0,-1) \hookrightarrow \pi^*\mathcal{V}_2 \twoheadrightarrow (V/\breve{W})_{\mathbf{P}}. \end{align*}$$

Returning to $\varphi $ , observe that the morphism of 3.5 factors through a morphism $S^\circ \to \mathbf {P}^\circ $ given by

$$\begin{align*}[\ell] \mapsto \big(\ell \cap \mathbf{P}\breve{W} \mapsto \varphi([\ell]) \in \operatorname{\mathrm{proj}}_\infty(\ell)\big). \end{align*}$$

The image of this morphism can be described in terms of the geometry of , where is the plane in $\mathbf {P} V$ spanned by $\infty $ and a line $\ell _0 \subset \mathbf {P}\overline {V}$ .

Lemma 3.7. The image of $S^\circ \to \mathbf {P}^\circ $ is the scheme

If $(X,\infty ,\mathbf {P}\breve {W})$ satisfies 3.1(i), then the morphism $S^\circ \to T^\circ $ is quasi-finite of degree q.

Proof. A point $(y \mapsto y_0 \in \ell _0) \in \mathbf {P}^\circ $ lies in the image $T^\circ $ if and only if there is a line $\ell $ contained in through y. If $P_{\ell _0} \subset X$ , then any y is a vertex of the plane $X_{\ell _0}$ , and any line in $X_{\ell _0}$ through y and not passing through $\infty $ witnesses membership in $T^\circ $ . Otherwise, $X_{\ell _0}$ is a q-bic curve which contains the line

$$\begin{align*}\langle y,\infty \rangle = \operatorname{\mathrm{proj}}_\infty^{-1}(y_0) = \operatorname{\mathrm{proj}}_\infty^{-1}(\ell_0 \cap \mathbf{P}\overline{W}) = P_{\ell_0} \cap \mathbf{P}\breve{W} \end{align*}$$

spanned by y and $\infty $ as an irreducible component. Thus, $(y \mapsto y_0 \in \ell _0)$ is contained in $T^\circ $ if and only if the residual curve $X_{\ell _0} - \ell _{y,\infty }$ contains a line passing through y. Classification of q-bic curves, as in 2.4, shows that this happens if and only if $X_{\ell _0}$ is a cone with vertex y. Finally, if the cone situation satisfies 3.1(i), then only this second case occurs, and this analysis shows that the fibers of $S^\circ \to T^\circ $ are the length q schemes parameterizing the lines in $X_{\ell _0} - \ell _{y,\infty }$ .

3.8. Closure of $T^\circ $

To describe the closure of $T^\circ $ in $\mathbf {P}$ , first extend the moduli description in 3.7 to all of $\mathbf {P}$ by considering the locally free $\mathcal {O}_{\mathbf {P}}$ -module $\mathcal {P}$ of rank $3$ defined by the diagram

where the upper sequence is as in 3.6. The fiber at a point $(y \mapsto y_0 \in \ell _0) \in \mathbf {P}$ of

• • $\mathcal {P}$ is the subspace of V underlying the plane $P_{\ell _0}$ ;

• • $\pi ^*\mathcal {V}_1$ is the line ;

• • the tautological subbundle $\mathcal {O}_{\mathbf {P}}(-1,0) \hookrightarrow \pi ^*\mathcal {V}_1$ is the point y.

Let $\beta _{\mathcal {P}} \colon \mathcal {P}^{[1]} \otimes \mathcal {P} \to \mathcal {O}_{\mathbf {P}}$ be the restriction of the q-bic form $\beta $ . Since $\mathcal {V}_1$ is isotropic—it parameterizes lines in X!—the adjoints of $\beta _{\mathcal {P}}$ induce maps

$$\begin{align*}\beta^\vee_{\mathcal{P}} \colon \pi^*\mathcal{V}_1^{[1]} \xrightarrow{\beta^\vee\rvert_{\mathcal{V}_1}} \pi^*\mathcal{V}_2^\vee \xrightarrow{\mathrm{eu}_{\pi_2}^\vee} \mathcal{O}_{\mathbf{P}}(0,1) \quad\text{and}\quad \beta_{\mathcal{P}} \colon \pi^*\mathcal{V}_1 \xrightarrow{\beta\rvert_{\mathcal{V}_1}} \pi^*\mathcal{V}_2^{[1],\vee} \xrightarrow{\mathrm{eu}_{\pi_2}^{[1],\vee}} \mathcal{O}_{\mathbf{P}}(0,q). \end{align*}$$

Set and . Then vanishes at points where the line in $\mathcal {P}$ given by $\mathrm {eu}_{\pi _1}$ lies in the radical of $\beta _{\mathcal {P}}$ , or, geometrically:

In particular, $T^\circ = T' \cap \mathbf {P}^\circ $ .

This contains too much: for instance, $T'$ contains the intersection of the irreducible components of , consisting of $(\infty \mapsto y_0 \in \ell _0),$ where $\ell _0 \subset \mathbf {P}\overline {W}$ . In fact, the map $v = (v_1,v_2)^\vee $ often factors through . To explain, given splittings $\mathcal {V}_1 \cong \mathcal {O}_C(-1) \oplus L_C$ and $\mathcal {V}_2 \cong \mathcal {T} \oplus (V/\breve {W})_C$ , write

$$\begin{align*}u_1' \colon \mathcal{O}_{\mathbf{P}}(-1,0) \stackrel{\mathrm{eu}_{\pi_1}}{\longrightarrow} \pi^*\mathcal{V}_1 \longrightarrow L_{\mathbf{P}} \quad\text{and}\quad u_2' \colon \mathcal{O}_{\mathbf{P}}(0,-1) \stackrel{\mathrm{eu}_{\pi_2}}{\longrightarrow} \pi^*\mathcal{V}_2 \longrightarrow \pi^*\mathcal{T} \end{align*}$$

for the projection of the Euler sections to the subbundle.

Lemma 3.9. Assume there exists a two-dimensional subspace $U \subset V$ such that:

1. (i) $U \cap \breve {W} = L$ , and

2. (ii) U admits an orthogonal complement W in V.

Then the induced splittings $\mathcal {V}_1 \cong \mathcal {O}_C(-1) \oplus L_C$ and $\mathcal {V}_2 \cong \mathcal {T} \oplus (V/\breve {W})_C$ are such that the adjoint maps

are diagonal, and there exists a factorization $v = v' \circ (u_1,u_2)^\vee $ , where

$$\begin{align*}v' \colon \mathcal{O}_{\mathbf{P}}(1,0) \otimes \pi^*\mathcal{O}_C(-1) \oplus \mathcal{O}_{\mathbf{P}}(0,1) \otimes (V/\breve{W})_{\mathbf{P}} \to \mathcal{O}_{\mathbf{P}}(q,1) \oplus \mathcal{O}_{\mathbf{P}}(1,q) \end{align*}$$

is the map given by the matrix

$$\begin{align*}\begin{pmatrix} u_2' \cdot \beta_1^\vee \cdot u_1^{q-1} & \beta_2^\vee \cdot u_1^{\prime{q}} \\ u_2^{\prime{q}} \cdot \beta_1 & u_2^{\prime{q-1}} \cdot \beta_2 \cdot u_1' \end{pmatrix}. \end{align*}$$

Proof. The decomposition $V = W \oplus U$ induces splittings $\mathcal {V}_1 \cong \mathcal {O}_C(-1) \oplus L_C$ and $\mathcal {V}_2 \cong \mathcal {T} \oplus (V/\breve {W})_C$ by intersecting $\mathcal {V}_1$ with $W_C$ as subbundles of $V_C$ , and by projecting $U_C$ to $\mathcal {V}_2$ , respectively. The adjoint maps of $\beta $ are now diagonal since W and U are orthogonal complements. The Euler sections split as $\mathrm {eu}_{\pi _1} = (u_1,u_1')^\vee $ and $\mathrm {eu}_{\pi _2} = (u_2',u_2)^\vee $ , and so

$$ \begin{align*} v_1 & = \mathrm{eu}_{\pi_2}^\vee \cdot \beta^\vee\rvert_{\mathcal{V}_1} \cdot \mathrm{eu}_{\pi_1}^{[1]} = u_2' \cdot \beta_1^\vee \cdot u_1^q + u_2 \cdot \beta_2^\vee \cdot u_1^{\prime{q}},\;\text{and} \\ v_2 & = \mathrm{eu}_{\pi_2}^{[1],\vee} \cdot \beta\rvert_{\mathcal{V}_1} \cdot \mathrm{eu}_{\pi_1} = u_2^{\prime{q}} \cdot \beta_1 \cdot u_1 + u_2^q \cdot \beta_2 \cdot u_1'. \end{align*} $$

Since $u_1$ and $u_2$ are locally multiplication by a scalar, their action commutes with the others, and a direct computation shows that there is a factorization $v = v \circ (u_1,u_2)^\vee $ .

The hypotheses of 3.9 are satisfied, for example, in all of the cone situations 3.2(i)–3.2, so, in particular, in all smooth cone situations as in 3.3(ii): in 3.2(i), U may be taken to be the span of L with any other isotropic Hermitian vector not lying in $\breve {W}$ ; in 3.2 3.2 and 3.2 3.2, U is necessarily the subspace supporting the subform of type $\mathbf {N}_2$ . In contrast, no U exists in 3.2 (v).

This provides a candidate for the Zariski closure of $T^\circ $ in $\mathbf {P}$ . To state the result, write

and $\wedge u \colon \mathcal {E}_2 \to \det (\mathcal {E}_2)$ for the natural surjection to the torsion-free quotient of $\operatorname {\mathrm {coker}}(u \colon \mathcal {O}_{\mathbf {P}} \to \mathcal {E}_2)$ .

Proposition 3.10. In the setting of 3.9, the scheme contains the Zariski closure of $T^\circ $ in $\mathbf {P}$ and is the rank $1$ degeneracy locus of

If furthermore $\dim S^\circ = 2$ , then $\dim T = 2$ , T is connected, Cohen–Macaulay, and there is an exact complex of sheaves on $\mathbf {P}$ given by

$$\begin{align*}0 \longrightarrow \mathcal{E}_2(-q-1,-q-1) \stackrel{\phi}{\longrightarrow} \mathcal{E}_1(-q-1,-q-1) \stackrel{\wedge^2\phi^\vee}{\longrightarrow} \mathcal{O}_{\mathbf{P}} \longrightarrow \mathcal{O}_T \longrightarrow 0. \end{align*}$$

In many cases, T is the closure of $T^\circ $ . This is verified for smooth cone situations in 4.2.

3.11.

Write $\mathcal {T}_{\pi _i}$ for the pullback to $\mathbf {P}$ of the relative tangent bundle of $\pi _i \colon \mathbf {P}\mathcal {V}_i \to C$ , and set

where $\mathcal {H}$ extracts the homology sheaf, and $\mathcal {P}$ is as in 3.8. Its associated $\mathbf {P}^1$ -bundle is

Extracting the line $[\ell ]$ yields a birational morphism $\mathbf {P}\mathcal {V} \to \mathbf {G}$ which is an isomorphism away from the locus where either $\infty \in \ell $ or $\ell \subset \mathbf {P}\breve {W}$ . The discussion preceding 3.7 implies that S is contained in the image of $\mathbf {P}\mathcal {V}$ and is not completely contained in the non-isomorphism locus, so it has a well-defined strict transform $\tilde S$ along $\mathbf {P}\mathcal {V} \to \mathbf {G}$ that has a morphism $\tilde S \to T$ ; when $(X,\infty ,\mathbf {P}\breve {W})$ satisfies 3.1(i), this resolves the rational map $\varphi \colon S \dashrightarrow T$ from 3.5. In summary, there is a commutative diagram:

From now on, assume that $(X,\infty ,\mathbf {P}\breve {W})$ satisfies 3.1(i). The remainder of this section is dedicated to constructing equations for $\tilde {S}$ by describing it as a bundle of q-bic points over T. Write $\rho \colon \mathbf {P}\mathcal {V}_T \to T$ for the projection of $\mathbf {P}\mathcal {V} \to \mathbf {P}$ restricted to T. Then 3.7 implies that $\tilde {S}$ is a hypersurface in $\mathbf {P}\mathcal {V}_T$ . Let $(\mathcal {P}_T,\beta _{\mathcal {P}_T})$ be the restriction to T of the q-bic form $(\mathcal {P},\beta _{\mathcal {P}})$ from 3.8. As a first step, we have the following.

Lemma 3.12. The q-bic form $\beta _{\mathcal {P}_T}$ induces a q-bic form $\beta _{\mathcal {V}_T} \colon \mathcal {V}_T^{[1]} \otimes \mathcal {V}_T \to \mathcal {O}_T$ whose q-bic equation

$$\begin{align*}\beta_{\mathcal{V}_T}(\mathrm{eu}_\rho^{[1]}, \mathrm{eu}_\rho) \colon \mathcal{O}_\rho(-q-1) \hookrightarrow \rho^*\mathcal{V}_T^{[1]} \otimes \rho^*\mathcal{V}_T \rightarrow \mathcal{O}_{\mathbf{P}\mathcal{V}_T} \end{align*}$$

vanishes at $((y \in \ell ) \mapsto (y_0 \in \ell _0)) \in \mathbf {P}\mathcal {V}_T$ if and only if $\ell $ is an isotropic line for $\beta $ . In particular, this section vanishes on the strict transform $\tilde {S}$ of S.

3.13.

Comparing the homology sheaf construction of $\mathcal {V}$ from 3.11 with the sequences for $\mathcal {V}_1$ and $\mathcal {V}_2$ from 3.6 gives a canonical short exact sequence

$$\begin{align*}0 \to \mathcal{T}_{\pi_1}(-1,0) \to \mathcal{V} \to \mathcal{O}_{\mathbf{P}}(0,-1) \to 0. \end{align*}$$

The subbundle may be identified via the Euler sequence for $\mathbf {P}\mathcal {V}_1 \to C$ as

$$\begin{align*}\mathcal{T}_{\pi_1}(-1,0) = \operatorname{\mathrm{coker}}(\mathrm{eu}_{\pi_1} \colon \mathcal{O}_{\mathbf{P}}(-1,0) \to \pi^*\mathcal{V}_1) \cong \det(\pi^*\mathcal{V}_1)(1,0) \cong \mathcal{O}_{\mathbf{P}}(1,0) \otimes \pi^*\mathcal{O}_C(-1) \otimes L. \end{align*}$$

Its inverse image under the quotient map $\mathcal {P} \to \mathcal {V}$ is the subbundle $\pi ^*\mathcal {V}_1$ , so its points are

$$\begin{align*}\mathbf{P}(\mathcal{T}_{\pi_1}(-1,0)) = \{((y \in \ell) \mapsto (y_0 \in \ell_0)) \in \mathbf{P}\mathcal{V} : \ell = \langle y_0, \infty\rangle\}. \end{align*}$$

Since $\ell = \langle y_0,\infty \rangle \subset \mathbf {P}\breve {W}$ whenever $(y \mapsto y_0 \in \ell _0) \in T$ , this subbundle is isotropic for $\beta _{\mathcal {V}_T}$ by 3.12, yielding the following observation.

Lemma 3.14. The q-bic equation $\beta _{\mathcal {V}_T}(\mathrm {eu}_\rho ^{[1]}, \mathrm {eu}_\rho )$ from 3.12 vanishes on $\mathbf {P}(\mathcal {T}_{\pi _1}(-1,0)\rvert _T)$ , and so it factors through the section

where $u_3 \colon \mathcal {O}_\rho (-1) \to \rho ^*\mathcal {V}_T \to \rho ^*\mathcal {O}_T(0,-1)$ is the equation of the subbundle.

5.1.

Throughout Sections 5–7, let $(V,\beta )$ be a q-bic form of type $\mathbf {1}^{\oplus 3} \oplus \mathbf {N}_2$ , and let X be the associated q-bic threefold. The two kernels and of $\beta $ underlie the singular point $x_+$ and the special smooth point $x_-$ of X. Setting , the classification of q-bic forms provides a canonical orthogonal decomposition $V = W \oplus U,$ where $\beta _W$ is of type $\mathbf {1}^{\oplus 3}$ . The plane $\mathbf {P} W$ intersects X at the smooth q-bic curve C defined by $\beta _W$ ; this curve is canonically identified as the base of the cones $X_-$ and $X_+$ in the cone situations

$$\begin{align*}(X,x_-,\mathbf{P} L_-^{[1],\perp}) \quad\text{and}\quad (X,x_+,\mathbf{P} L_+^{\perp,[-1]}) \end{align*}$$

as in 3.2 (ii) and 3.2 (iii).

The Fano scheme S of lines in X is of expected dimension $2$ : this follows, for example, from 3.3(i) applied to either cone situation above. Alternatively, by 3.4(i) and 3.4(ii), the subschemes $C_\pm \subset S$ parameterizing lines through the special points $x_\pm \in X$ are supported on a scheme isomorphic to C, so the singular locus of S has dimension $1$ , and the discussion of 2.8 shows that S is a surface.

5.2. Automorphisms

Consider the automorphism group scheme ${\mathbf {Aut}}(V,\beta )$ of the q-bic form, as introduced in [Reference Cheng9, Section 5.1]. Specializing the computation of [Reference Cheng8, Proposition 1.3.7] with $a = 1$ and $b = 3$ gives the following explicit description as a closed sub-group scheme of $\mathbf {GL}_5$ :

A particularly useful subgroup is that which preserves the orthogonal decomposition $V = W \oplus U$ :

Observe that the torus $\mathbf {G}_m$ in ${\mathbf {Aut}}(V,\beta )$ acts, in particular, on X and the Fano surface S; it is straightforward to check that its fixed schemes are $X^{\mathbf {G}_m} = \{x_-, x_+\} \cup C$ and $S^{\mathbf {G}_m} = C_- \cup C_+$ , respectively.

5.3. Cone situations

Both cone situations in 5.1 associated with the special points $x_\mp \in X$ satisfy 3.1(i) and 3.1(iii), so 3.5 gives rational maps

$$\begin{align*}\varphi_- \colon S \longrightarrow C \quad\text{and}\quad \varphi_+ \colon S \dashrightarrow C, \end{align*}$$

where the orthogonal decomposition $V = W \oplus U$ identifies the target as the curve $C = X \cap \mathbf {P} W$ . Since $\varphi _-$ arises from a smooth cone situation, it extends to a morphism by 4.9, whereas $\varphi _+$ is defined only away from $C_+$ . Writing $\operatorname {\mathrm {proj}}_{\mathbf {P} U} \colon X \dashrightarrow \mathbf {P} W$ for the rational map induced by linear projection centered at $\mathbf {P} U$ , the specific geometry of X offers an alternative description of the maps $\varphi _\mp $ .

Lemma 5.4. Let $\ell \subset X$ be a line not passing through either $x_\mp $ . Then is a line in $\mathbf {P} W$ tangent to C at the point $\operatorname {\mathrm {proj}}_{\mathbf {P} U}(\ell \cap X_+)$ with residual intersection point $\operatorname {\mathrm {proj}}_{\mathbf {P} U}(\ell \cap X_-)$ , so

$$ \begin{align*} \varphi_+([\ell]) & = \text{point of tangency between } \ell_0 \text{ and } C, \\ \varphi_-([\ell]) & = \text{residual point of intersection between } \ell_0 \text{ and } C. \end{align*} $$

Proof. Projection from $\mathbf {P} U$ contracts only lines through $x_\mp $ , so $\ell _0$ is a line in $\mathbf {P} W$ . Since $\mathbf {P} U$ intersects X only at the vertices $x_\mp $ of the two cones over C, and since $x_+$ is a singular point of multiplicity q,

$$\begin{align*}C \cap \ell_0 = \operatorname{\mathrm{proj}}_{\mathbf{P} U}\big(\ell \cap \operatorname{\mathrm{proj}}_{\mathbf{P} U}^{-1}(C)\big) = \operatorname{\mathrm{proj}}_{\mathbf{P} U}\big(\ell \cap (X_- \cup q X_+)\big). \end{align*}$$

Thus, $\ell _0$ and C are tangent at $\operatorname {\mathrm {proj}}_{\mathbf {P} U}(\ell \cap X_+)$ and have residual point of intersection at $\operatorname {\mathrm {proj}}_{\mathbf {P} U}(\ell \cap X_-)$ . On the other hand, the definition of $\varphi _\mp $ from 3.5 gives

$$\begin{align*}\varphi_\mp([\ell]) = \operatorname{\mathrm{proj}}_{x_\mp}(\ell \cap X_\mp) = \operatorname{\mathrm{proj}}_{\mathbf{P} U}(\ell \cap X_\mp) \end{align*}$$

with the second equality because $x_\pm \notin X_\mp $ .

With more notation, the proof works for families of lines in . This leads to another proof of 4.9, that $\varphi _-$ extends over $C_-$ : 5.4 means that $\varphi _-([\ell ])$ is the residual intersection point of C with its tangent line at $\operatorname {\mathrm {proj}}_{\mathbf {P} U}(\ell \cap X_+)$ , and this makes sense even when $x_- \in \ell $ . More interestingly, this gives an alternative moduli interpretation of as a scheme over C via $\varphi _+$ . To describe it, let $\mathcal {E}_C$ be the embedded tangent bundle C in $\mathbf {P} W$ , which is defined via pullback of the right square in the following commutative diagram of short exact sequences:

where, as in 3.6, . The fiber of $\mathcal {E}_C$ over a point $y_0 = \mathbf {P} L_0$ is the subspace $L_0^{[1],\perp }$ in W underlying the embedded tangent space $\mathbf {T}_{C,y_0}$ (see 2.5). The subbundle of $V_C$ defines the family $\mathbf {P}\mathcal {W} \to C$ of hyperplanes in $\mathbf {P} V$ spanned by the tangent lines to C and U. Let be the corresponding family of hyperplane sections of X. A reformulation of 5.4 in these terms is as follows.

Corollary 5.5. Let $\mathbf {L}^\circ $ be the restriction of the Fano correspondence $S \leftarrow \mathbf {L} \rightarrow X$ to . Then the morphism $\mathbf {L}^\circ \to X$ factors through $X_{\mathbf {P}\mathcal {W}}$ and fits into a commutative diagram

A line such that $\varphi _+([\ell ]) = y_0$ intersects $X_+$ along the line $\langle x_+, y_0 \rangle $ . These lines are globally parameterized by the subbundle ; together with 5.4 and 5.5, this implies that $\mathbf {L}^\circ \times _{X_{\mathbf {P}\mathcal {W}}} \mathbf {P}\mathcal {W}'$ projects isomorphically to . This suggests that $\varphi _+$ may be resolved by considering all lines in $X_{\mathbf {P}\mathcal {W}}$ that pass through $\mathbf {P}\mathcal {W}'$ . To proceed, consider the morphism $X_{\mathbf {P}\mathcal {W}} \to C$ .

Proof. This family of q-bic surfaces over C is determined by the q-bic form $\beta _{\mathcal {W}} \colon \mathcal {W}^{[1]} \otimes \mathcal {W} \to \mathcal {O}_C$ obtained by restricting $\beta $ to $\mathcal {W}$ . The orthogonal splitting $V = W \oplus U$ restricts to a decomposition

$$\begin{align*}(\mathcal{W},\beta_{\mathcal{W}}) = (\mathcal{E}_C, \beta_{W,\mathrm{tan}}) \oplus (U_C, \beta_U), \end{align*}$$

where $\beta _{W,\mathrm {tan}}$ is q-bic form induced by $\beta _W$ on the embedded tangent spaces to C. Observe that $\beta _U$ is of constant type $\mathbf {N}_2$ , and $\beta _{W,\mathrm {tan}}$ is of type $\mathbf {0} \oplus \mathbf {1}$ or $\mathbf {N}_2$ depending on whether or not the point of tangency is a Hermitian point of C; in particular, $\beta _{\mathcal {W}}$ is everywhere of corank $2$ . By [Reference Cheng10, Corollary 2.6], the nonsmooth locus of $X_{\mathbf {P}\mathcal {W}} \to C$ corresponds to the subbundle

$$\begin{align*}\mathcal{W}^\perp = \mathcal{E}_C^{\perp_{\beta_{W,\mathrm{tan}}}} \oplus U_C^{\perp_{\beta_U}} = \mathcal{O}_C(-q) \oplus L_{+,C}^{[1]} \subset \mathcal{W}^{[1]}, \end{align*}$$

where the two orthogonals can be computed geometrically by noting that $X \cap \mathbf {P} U$ is singular at $x_+$ , and the intersection between C and its tangent line at $y_0$ is singular at $y_0$ . This bundle descends through Frobenius to $\mathcal {W}'$ as in the statement, and so $X_{\mathbf {P}\mathcal {W}}$ is singular thereon.

Construct the family of lines in $X_{\mathbf {P}\mathcal {W}}$ over C through $\mathbf {P}\mathcal {W}'$ as follows: Set , let be the associated $\mathbf {P}^1$ -bundle, and write $\tilde \varphi _+ \colon S^\nu \to C$ for the structure map. Linear projection over C with center $\mathbf {P}\mathcal {W}'$ produces a rational map $X_{\mathbf {P}\mathcal {W}} \dashrightarrow S^\nu $ which is resolved on the blowup $\tilde {X}_{\mathbf {P}\mathcal {W}}$ along $\mathbf {P}\mathcal {W}'$ . Since $X_{\mathbf {P}\mathcal {W}}$ is a family of q-bic surfaces over C, its singular locus has multiplicity q, the fibers of $\tilde {X}_{\mathbf {P}\mathcal {W}} \to S^\nu $ are curves of degree $1$ in $\mathbf {P} V$ . In other words, this is a family of lines in X, and so defines a morphism $\nu \colon S^\nu \to S$ .

Proposition 5.7. The morphism $\nu \colon S^\nu \to S$ is the normalization, fits into a commutative diagram

and satisfies $\nu ^*\mathcal {O}_S(1) = \tilde \varphi _+^*\mathcal {O}_C(1) \otimes \mathcal {O}_{\tilde \varphi _+}(q+1) \otimes L_+^\vee $ .

Proof. Note $\mathcal {W}" \cong \mathcal {T}_C(-1) \oplus L_{-,C}$ and the fibers of $\tilde {X}_{\mathbf {P}\mathcal {W}} \to S^\nu $ over are those lines through $x_+ \in X$ . The statement of and the comments following the corollary 5.5 imply that and represent the same moduli problem over C, and this implies the first two statements.

To compute the pullback of the Plücker line bundle, describe the $\mathbf {P}^1$ -bundle $\tilde {X}_{\mathbf {P}\mathcal {W}} \to S^\nu $ more precisely: The blowup of $\mathbf {P}\mathcal {W}$ along $\mathbf {P}\mathcal {W}'$ is canonically the $\mathbf {P}^2$ -bundle over $S^\nu $ associated with $\tilde {\mathcal {W}}$ formed in the pullback diagram

and so the exceptional divisor of $\mathbf {P}\tilde {\mathcal {W}} \to \mathbf {P}\mathcal {W}$ is the subbundle $\mathbf {P}(\tilde \varphi _+^*\mathcal {W}') \subset \mathbf {P}\tilde {\mathcal {W}}$ . The inverse image of $X_{\mathbf {P}\mathcal {W}}$ along this blowup is the bundle of q-bic curves over $S^\nu $ defined by the q-bic form $ \beta _{\tilde {\mathcal {W}}} \colon \tilde {\mathcal {W}}^{[1]} \otimes \tilde {\mathcal {W}} \to \mathcal {O}_{S^\nu } $ obtained by restricting $\tilde \varphi ^*_+\beta _{\mathcal {W}}$ . Since $\mathcal {W}^\perp = \mathcal {W}^{\prime [1]}$ by 5.6, the diagram above implies that $\tilde \varphi _+^*\mathcal {W}^{\prime [1]} = \tilde {\mathcal {W}}^\perp $ . Thus, by [Reference Cheng9, Lemma 1.5], there is an exact sequence

$$\begin{align*}0 \to \mathcal{K} \to \tilde{\mathcal{W}} \xrightarrow{\beta_{\tilde{\mathcal{W}}}} \tilde{\mathcal{W}}^{[1],\vee} \to \tilde\varphi_+^*\mathcal{W}^{\prime[1],\vee} \to 0, \end{align*}$$

where $\mathcal {K}$ is the rank $2$ subbundle defining the $\mathbf {P}^1$ -bundle $\tilde {X}_{\mathbf {P}\mathcal {W}} \to S^\nu $ . So since $\nu ^*\mathcal {S} = \mathcal {K}$ ,

$$ \begin{align*} \nu^*\mathcal{O}_S(1) \cong \nu^*\det(\mathcal{S})^\vee \cong \det(\mathcal{K})^\vee & \cong \det(\tilde{\mathcal{W}})^{\vee,\otimes q+1} \otimes \tilde\varphi^*_+\det(\mathcal{W}')^{\otimes q} \\ & \cong \tilde\varphi^*_+\det(\mathcal{W}')^\vee \otimes \mathcal{O}_{\tilde\varphi_+}(q+1) \\ & \cong \tilde\varphi^*_+\mathcal{O}_C(1) \otimes \mathcal{O}_{\tilde\varphi_+}(q+1) \otimes L_+^\vee \end{align*} $$

since $\mathcal {W}' = \mathcal {O}_C(-1) \oplus L_{+,C}$ .

The following statement summarizes 5.4 and 5.7 and gives a direct geometric relationship between the maps $\tilde \varphi _+$ and $\varphi _-$ . For this, let $\phi _C \colon C \to C$ be the endomorphism that sends a point x to the residual intersection point between C and its tangent line at x, as described in 2.6.

Corollary 5.8. There is a commutative diagram of morphisms

5.9. Conductors

Consider the conductor ideals associated with the normalization $\nu \colon S^\nu \to S$ :

The corresponding conductor subschemes $D \subset S$ and $D^\nu \subset S^\nu $ are thickenings of the curves $C_+$ and $C_+^\nu $ , respectively, and fit into a commutative diagram

where the vertical maps are finite by properness of S and $S^\nu $ over C. Algebraically, the conductor ideal of S is characterized as the largest ideal of $\mathcal {O}_S$ which is also an ideal of $\nu _*\mathcal {O}_{S^\nu }$ , so there is a commutative diagram of exact sequences of sheaves on S:

Duality theory for $\nu \colon S^\nu \to S$ identifies the conductor ideal $\operatorname {cond}_{\nu ,S^\nu }$ with the relative dualizing sheaf $\omega _{S^\nu /S} \cong \omega _{S^\nu } \otimes \nu ^*\omega _S^\vee $ , compare [54, Tag 0FKW] and [Reference Reid42, Proposition 2.3].

Proposition 5.10. The conductor ideal of $S^\nu $ is isomorphic to

In particular, the conductor subscheme $D^\nu $ is the $\delta $ -order neighborhood of $C_+^\nu $ .

Proof. Since $S^\nu $ is the $\mathbf {P}^1$ -bundle on $\mathcal {W}" \cong \mathcal {T}_C(-1) \oplus L_{-,C}$ , as in 5.7, the relative Euler sequence gives

$$\begin{align*}\omega_{S^\nu} \cong \omega_{S^\nu/C} \otimes \tilde\varphi_+^*\omega_C \cong \mathcal{O}_{\tilde\varphi_+}(-2) \otimes \tilde\varphi^*_+(\omega_C^{\otimes 2} \otimes \mathcal{O}_C(1)) \otimes L_-^\vee. \end{align*}$$

By 2.8, $\omega _S \cong \mathcal {O}_S(2q - 3) \otimes (L_+\otimes L_-)^{\vee ,\otimes 2}$ , so 5.7 gives

$$ \begin{align*} \nu^*\omega_S & \cong \nu^*\mathcal{O}_S(2q-3) \otimes (L_+ \otimes L_-)^{\vee, \otimes 2} \\ & \cong \tilde\varphi_+^*\mathcal{O}_C(2q-3) \otimes \mathcal{O}_{\tilde\varphi_+}\big((2q-3)(q+1)\big) \otimes L_+^{\vee, \otimes 2q-1} \otimes L_-^{\vee,\otimes 2}. \end{align*} $$

Since C is a plane curve of degree $q+1$ , $\omega _C^{\otimes 2} \otimes \mathcal {O}_C(1) \cong \mathcal {O}_C(2q-3)$ , and so

$$\begin{align*}\omega_{S^\nu/S} \cong \omega_{S^\nu} \otimes \nu^*\omega_S^\vee \cong \mathcal{O}_{\tilde\varphi_+}(-\delta-1) \otimes (L_+^{\otimes 2q-1} \otimes L_-). \end{align*}$$

5.11. The sheaf $\mathcal {F}$

Let $\mathcal {D}$ and $\mathcal {D}^\nu $ be the coherent $\mathcal {O}_C$ -algebras by pushing structure sheaves along the finite morphisms $\varphi _- \colon D \to C$ and $\phi _C \circ \tilde \varphi _+ \colon D^\nu \to C$ , respectively. Restricting $\nu $ to the conductors induces an injective map $\nu ^\# \colon \mathcal {D} \to \mathcal {D}^\nu $ of $\mathcal {O}_C$ -algebras. Its cokernel

identifications arising from the diagram of 5.9, is an $\mathcal {O}_C$ -module related to S as follows.

Lemma 5.12. There is an exact sequence of finite locally free $\mathcal {O}_C$ -modules

$$\begin{align*}0 \to \mathcal{O}_C \to \phi_{C,*} \mathcal{O}_C \to \mathcal{F} \to \mathbf{R}^1\varphi_{-,*}\mathcal{O}_S \to 0 \end{align*}$$

in which the map $\mathcal {F} \to \mathbf {R}^1\varphi _{-,*}\mathcal {O}_S$ splits.

The importance of $\mathcal {F}$ is in the following relation with the cohomology of the structure sheaf of S.

Proposition 5.13. The cohomology of $\mathcal {O}_S$ is given by

$$\begin{align*}\mathrm{H}^i(S,\mathcal{O}_S) \cong \begin{cases} \mathrm{H}^0(C,\mathcal{O}_C) & \text{if}\; i = 0, \\ \mathrm{H}^0(C,\mathcal{F}) & \text{if}\; i = 1,\;\text{and} \\ \mathrm{H}^1(C,\mathcal{F})/\mathrm{H}^1(C,\mathcal{O}_C) & \text{if}\; i = 2. \end{cases} \end{align*}$$

Proof. Consider the cohomology sequence associated with the exact sequence

$$\begin{align*}0 \to \mathcal{O}_S \to \nu_*\mathcal{O}_{S^\nu} \to \nu_*\mathcal{O}_{S^\nu}/\mathcal{O}_S \to 0. \end{align*}$$

Note that $\varphi _{-,*}\mathcal {O}_S \cong \mathcal {O}_C$ by 4.10 and, for each $i = 0,1,2$ ,

$$\begin{align*}\mathrm{H}^i(S,\nu_*\mathcal{O}_{S^\nu}/\mathcal{O}_S) \cong \mathrm{H}^i(C,\mathcal{F}) \quad\text{and}\quad \mathrm{H}^i(S^\nu,\mathcal{O}_{S^\nu}) \cong \mathrm{H}^i(C,\mathcal{O}_C) \end{align*}$$

since $\nu _*\mathcal {O}_{S^\nu }/\mathcal {O}_S$ is supported on D and $D \to C$ is affine, and the fact that $\tilde \varphi _+\colon S^\nu \to C$ is a projective bundle. Thus, $\mathrm {H}^0(S,\mathcal {O}_S) \cong \mathrm {H}^0(S^\nu ,\mathcal {O}_{S^\nu }) \cong \mathrm {H}^0(C,\mathcal {O}_C)$ and there is an exact sequence

$$\begin{align*}0 \to \mathrm{H}^0(C,\mathcal{F}) \xrightarrow{a} \mathrm{H}^1(S,\mathcal{O}_S) \xrightarrow{b} \mathrm{H}^1(S,\nu_*\mathcal{O}_{S^\nu}) \to \mathrm{H}^1(C,\mathcal{F}) \to \mathrm{H}^2(S,\mathcal{O}_S) \to 0. \end{align*}$$

The result will follow upon verifying that $b \colon \mathrm {H}^1(S,\mathcal {O}_S) \to \mathrm {H}^1(S,\nu _*\mathcal {O}_{S^\nu })$ vanishes. Since $\varphi _{-,*}\mathcal {O}_S \cong \mathcal {O}_C$ , the Leray spectral sequence gives a short exact sequence

$$\begin{align*}0 \to \mathrm{H}^1(C,\mathcal{O}_C) \xrightarrow{c} \mathrm{H}^1(S,\mathcal{O}_S) \xrightarrow{d} \mathrm{H}^0(C,\mathbf{R}^1\varphi_{-,*}\mathcal{O}_S) \to 0. \end{align*}$$

The splitting in 5.12 implies that the composite $ d \circ a \colon \mathrm {H}^0(C,\mathcal {F}) \to \mathrm {H}^0(C,\mathbf {R}^1\varphi _{-,*}\mathcal {O}_S) $ is a surjection. So the exactness of the long sequence means it remains to show that $ b \circ c \colon \mathrm {H}^1(C,\mathcal {O}_C) \to \mathrm {H}^1(S,\nu _*\mathcal {O}_{S^\nu }) $ vanishes. Pushing down to C along $\varphi _-$ and applying 5.8 shows that this is

$$\begin{align*}\phi_C \colon \mathrm{H}^1(C,\mathcal{O}_C) \to \mathrm{H}^1(C,\phi_{C,*}\mathcal{O}_C) \end{align*}$$

which, as in 5.12, is the map induced by the $q^2$ -power Frobenius up to an automorphism, and this is the zero map by 2.2.

Structure sheaf cohomology of S is therefore reduced to that of $\mathcal {F}$ , which will be computed when $q = p$ in 7.12. Combined with the Euler characteristic computation in 2.13, this gives Theorem B. In turn, cohomology of $\mathcal {F}$ is determined by describing its structure, and this is achieved in the next section via the following duality relationship with the algebra $\mathcal {D}$ .

Proposition 5.14. There is a canonical isomorphism of graded $\mathcal {O}_C$ -modules

$$\begin{align*}\mathcal{F} \cong \mathcal{D}^\vee \otimes \mathcal{O}_C(-q+1) \otimes L_+^{\otimes 2q-1} \otimes L_-^{\otimes 2}. \end{align*}$$

Proof. Applying $\mathbf {R}\mathcal {H}\!\mathit {om}_{\mathcal {O}_S}(-,\mathcal {O}_S)$ to the ideal sheaf sequence of the conductor $D \hookrightarrow S$ yields a triangle in the derived category of S

$$\begin{align*}\mathbf{R}\mathcal{H}\!\mathit{om}_{\mathcal{O}_S}(\mathcal{O}_D,\mathcal{O}_S) \to \mathcal{O}_S \to \nu_*\mathcal{O}_{S^\nu} \xrightarrow{+1} \end{align*}$$

since $\mathbf {R}\mathcal {H}\!\mathit {om}_{\mathcal {O}_S}(\mathcal {O}_S,\mathcal {O}_S) = \mathcal {O}_S$ and $\mathbf {R}\mathcal {H}\!\mathit {om}_{\mathcal {O}_S}(\operatorname {cond}_{\nu ,S},\mathcal {O}_S) = \nu _*\mathcal {O}_{S^\nu }$ , upon identifying $\operatorname {cond}_{\nu ,S}$ with $\nu _*\omega _{S^\nu /S}$ as in 5.9 and using duality for $\nu $ . The map $\mathcal {O}_S \to \nu _*\mathcal {O}_{S^\nu }$ is dual to evaluation at $1$ , and hence is the $\mathcal {O}_S$ -module map determined by $1 \mapsto 1$ ; in other words, this is the map $\nu ^\#$ , and so

$$\begin{align*}\nu_*\mathcal{O}_{S^\nu}/\mathcal{O}_S \cong \mathbf{R}\mathcal{H}\!\mathit{om}_{\mathcal{O}_S}(\mathcal{O}_D,\mathcal{O}_S)[1] \end{align*}$$

in the derived category of S. Applying $\mathbf {R}\varphi _{-,*}$ and applying relative duality for $\varphi _- \colon S \to C$ yields

$$\begin{align*}\mathcal{F} \cong \mathbf{R}\varphi_{-,*}\mathbf{R}\mathcal{H}\!\mathit{om}_{\mathcal{O}_S}(\mathcal{O}_D,\mathcal{O}_S)[1] \cong \mathbf{R}\mathcal{H}\!\mathit{om}_{\mathcal{O}_C}(\mathbf{R}\varphi_{-,*}(\mathcal{O}_D \otimes \omega_{\varphi_-}), \mathcal{O}_C)[1], \end{align*}$$

where $\omega _{\varphi _-} = (\omega _S \otimes \varphi _-^*\omega _C^\vee )[1]$ . By 2.8,

$$\begin{align*}\mathcal{O}_D \otimes \omega_S \cong \mathcal{O}_S(2q-3)\rvert_D \otimes L_+^{\vee, \otimes 2} \otimes L_-^{\vee, \otimes 2} \cong \varphi_-^*\mathcal{O}_C(2q-3)\rvert_D \otimes L_+^{\vee,\otimes 2q-1} \otimes L_-^{\vee, \otimes 2}, \end{align*}$$

where $\mathcal {O}_S(1)\rvert _D = \varphi _-^*\mathcal {O}_C(1)\rvert _D \otimes L_+^\vee $ since $\varphi _- \colon D \to C$ is induced by the line subbundle of $\mathcal {S}\rvert _D$ obtained by intersecting with $(L_- \oplus W)_D$ by 3.5, so there is a short exact sequence

$$\begin{align*}0 \to \varphi_-^*\mathcal{O}_C(-1)\rvert_D \to \mathcal{S}\rvert_D \to L_{+,D} \to 0, \end{align*}$$

and taking determinants yields the desired identification. Combining with $\omega _C \cong \mathcal {O}_C(q-2)$ gives

$$\begin{align*}\mathbf{R}\varphi_{-,*}(\mathcal{O}_D \otimes \omega_{\varphi_-}) = (\mathbf{R}\varphi_{-,*}\mathcal{O}_D) \otimes \mathcal{O}_C(q-1) \otimes L_+^{\vee,\otimes 2q-1} \otimes L_-^{\vee,\otimes 2}[1]. \end{align*}$$

Since $D \to C$ is of relative dimension $0$ , $\mathbf {R}\varphi _{-,*}\mathcal {O}_D = \varphi _{-,*}\mathcal {O}_D = \mathcal {D}$ , yielding the result.

5.15. The algebra $\mathcal {D}^\nu $

Before turning to $\mathcal {D}$ , consider the simpler algebra $\mathcal {D}^\nu $ associated with $\phi _C \circ \tilde \varphi _+ \colon D^\nu \to C$ . Since $D^\nu $ is disjoint from the curve $C_-^\nu = \mathbf {P} L_{-,C}$ parameterizing lines through $x_- \in X$ , it is contained in the $\mathbf {A}^1$ -bundle over C given by

the identification obtained by taking graphs of a linear function $\mathcal {T}_C(-1) \to L_{-,C}$ . Then 5.10 means that $D^\nu $ is the $\delta $ -order neighborhood of the zero section $C_+^\nu $ , so this identifies the graded $\mathcal {O}_C$ -module underlying the algebra of $D^\nu $ as

$$\begin{align*}\mathcal{D}^\nu = (\phi_C \circ \tilde\varphi_+)*\mathcal{O}_{D^\nu} \cong \phi_{C,*}\big(\bigoplus\nolimits_{i = 0}^\delta (\mathcal{T}_C(-1) \otimes L_-^\vee)^{\otimes i}\big). \end{align*}$$

6.2. Affine bundles

The duality relation 5.14 relates $\mathcal {F}$ with $\mathcal {D}$ , and the latter is a quotient of coordinate rings of schemes affine over C: namely, and $T^\circ $ as in 3.7 with respect to the smooth cone situation $(X,x_-,\mathbf {P} L_-^{[1],\perp })$ . These lie in affine space bundles

with $\mathcal {V}_1$ and $\mathcal {V}_2$ as in 3.6, and $\mathcal {V}$ is as in 3.11; set . Comparing the diagram in 3.6 with the description of the boundaries from 4.1 implies that there is a commutative diagram of affine schemes over C given by

Observe that the relative Euler sequences for these affine bundles give canonical isomorphisms

$$ \begin{align*} \mathcal{O}_{\mathbf{A}}(-1,0) & \cong \pi^*\mathcal{O}_C(-1), & \mathcal{T}_{\pi_2}(0,-1)\rvert_{\mathbf{A}} & \cong \pi^*\mathcal{T}, & \mathcal{O}_\rho(-1) & \cong \rho^*\mathcal{O}_{\mathbf{A}}(0,-1) \cong L_{+,\mathbf{B}}, \\ \mathcal{T}_{\pi_1}(-1,0)\rvert_{\mathbf{A}} & \cong L_{-,\mathbf{A}}, & \mathcal{O}_{\mathbf{A}}(0,-1) & \cong L_{+,\mathbf{A}}, & \mathcal{T}_\rho(-1)\rvert_{\mathbf{B}} & \cong \rho^*\mathcal{T}_{\pi_1}(-1,0)\rvert_{\mathbf{B}} \cong L_{-,\mathbf{B}}. \end{align*} $$

This identifies the $\mathcal {O}_C$ -algebras and as follows.

Lemma 6.3. The q-bic form $\beta $ induces an isomorphism

$$\begin{align*}\mathcal{A} \cong \operatorname{\mathrm{Sym}}^*(\mathcal{O}_C(-1) \otimes L_-^\vee) \otimes \operatorname{\mathrm{Sym}}^*(\Omega_{\mathbf{P} W}^1(1)\rvert_C \otimes L_+), \end{align*}$$

and endows the $\mathcal {A}$ -algebra $\mathcal {B}$ with an increasing filtration whose associated graded pieces are

Proof. The splittings $\mathcal {V}_1 \cong \mathcal {O}_C(-1) \oplus L_{-,C}$ and $\mathcal {V}_2 \cong \mathcal {T} \oplus L_{+,C}$ as in Section 4 give canonical relative projective coordinates on $\mathbf {P}\mathcal {V}_i$ over C, and identify the associated affine bundles as

$$\begin{align*}\mathbf{A} \cong \mathbf{A}(\mathcal{O}_C(1) \otimes L_-) \times_C \mathbf{A}(\mathcal{T} \otimes L_+^\vee). \end{align*}$$

This identifies $\mathcal {A} = \pi _*\mathcal {O}_{\mathbf {A}}$ as claimed.

For $\mathcal {B}$ , begin with the affine bundle $\rho \colon \mathbf {B} \to \mathbf {A}$ obtained as the complement of $\mathbf {P}(\mathcal {T}_{\pi _1}(-1,0))$ in $\mathbf {P}\mathcal {V}$ over $\mathbf {A}$ . Dualizing the short exact sequence in 3.13 and restricting to $\mathbf {A}$ gives a sequence

$$\begin{align*}0 \to \mathcal{O}_{\mathbf{A}}(0,1) \to \mathcal{V}^\vee\rvert_{\mathbf{A}} \to \Omega^1_{\pi_1}(1,0)\rvert_{\mathbf{A}} \to 0. \end{align*}$$

View $\mathcal {V}^\vee \rvert _{\mathbf {A}}$ as the linear functions on $\mathbf {P}\mathcal {V}\rvert _{\mathbf {A}}$ over $\mathcal {A}$ . A local generator for the subbundle $\mathcal {O}_{\mathbf {A}}(0,1)$ is then a linear equation defining $\mathbf {P}(\mathcal {T}_{\pi _1}(-1,0)\rvert _{\mathbf {A}})$ , and so becomes invertible on $\mathbf {B}$ . Therefore,

$$\begin{align*}\rho_*\mathcal{O}_{\mathbf{B}} \cong \operatorname{\mathrm{colim}}_n \operatorname{\mathrm{Sym}}^n(\mathcal{V}^\vee(0,-1))\rvert_{\mathbf{A,}} \end{align*}$$

where the transition maps are induced by multiplication by a local generator for the subbundle $\mathcal {O}_{\mathbf {A}} \hookrightarrow \mathcal {V}^\vee (0,-1)\rvert _{\mathbf {A}}$ . The $2$ -step filtration on $\mathcal {V}^\vee (0,-1)\rvert _{\mathbf {A}}$ starting with $\mathcal {O}_{\mathbf {A}}$ and followed by the entire bundle induces filtrations on the symmetric powers, compatible with the transition maps, whence a filtration on $\rho _*\mathcal {O}_{\mathbf {B}}$ with graded pieces

$$\begin{align*}\operatorname{\mathrm{gr}}_i\rho_*\mathcal{O}_{\mathbf{B}} \cong \Omega^1_{\pi_1}(1,-1)\rvert_{\mathbf{A}}^{\otimes i} \cong \mathcal{O}_{\mathbf{A}} \otimes (L_-^\vee \otimes L_+)^{\otimes i} \;\;\text{for all } i \in \mathbf{Z}_{\geq 0} \end{align*}$$

upon applying the identifications of 6.2. Pushing along $\pi \colon \mathbf {A} \to C$ then gives the result.