Generic Lines in Projective Space and the Koszul Property

In this paper, we study the Koszul property of the homogeneous coordinate ring of a generic collection of lines in $\mathbb{P}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in $\mathbb{P}^n.$ We show that if $\mathcal{M}$ is a collection of $m$ lines in general linear position in $\mathbb{P}^n$ with $2m \leq n+1$ and $R$ is the coordinate ring of $\mathcal{M},$ then $R$ is Koszul. Further, if $\mathcal{M}$ is a generic collection of $m$ lines in $\mathbb{P}^n$ and $R$ is the coordinate ring of $\mathcal{M}$ with $m$ even and $m +1\leq n$ or $m$ is odd and $m +2\leq n,$ then $R$ is Koszul. Lastly, we show if $\mathcal{M}$ is a generic collection of $m$ lines such that \[ m>\frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\] then $R$ is not Koszul. We give a complete characterization of the Koszul property of the coordinate ring of a generic collection of lines for $n \leq 6$ or $m \leq 6$. We also determine the Castelnuovo-Mumford regularity of the coordinate ring for a generic collection of lines and the projective dimension of the coordinate ring of collection of lines in general linear position.


CHAPTER 2. PRELIMINARIES
In this chapter we review some geometric concepts in projective space and review some of the necessary algebra to understand the results in Chapter 4.

Projective Space
Definition 2.1.1. Given a C−vector space V of dimension n+1, the projective space P n , or PV, is the set of equivalence classes of V − {0} under the equivalence relation ∼ defined by x ∼ y if and only if there is a non-zero element λ ∈ C such that x = λy. A point of P n is denoted as [x 0 : · · · : x n ], where x i ∈ C for every i. A projective variety V is a subset of P n that is the zero-locus of a family of homogeneous polynomials in n + 1 variables with coefficients in C. A projective variety is called irreducible if it cannot be expressed as a proper union of two projective varieties.
The varieties we will be working with are not irreducible, but they are unions of irreducible varieties.
Every projective variety can be studied by passing to an ideal in C[x 0 , . . . , x n ] and studying its corresponding quotient ring.
The set I(X ) is an ideal called the defining ideal of X and the corresponding quotient ring R = C[x 0 , . . . , x n ]/I(X ) is called the homogeneous coordinate ring of X .
A projective subspace PW of P n is of the form π (W − {0}) , where π is the residue class map and W is a subspace of V. Define dim(PW ) = dim(W ) − 1. Consider the following example of a projective variety and its homogeneous coordinate ring.
Example 2.1.3. In projective space a line is a subspace of dimension 1. Let L 1 and L 2 be the following two collections of points in P 3 defined as Thus, the coordinate ring is We digress momentarily to define an important geometric concept that we will use heavily.
Definition 2.1.4. Let P be a collection of p points in P n and M be a collection of m lines in P n .
The points of P are in general linear position if any s points span a P r , where r = min{s − 1, n}.
Similarly, the lines of M are in general linear position if any s lines span a P r , where r = min{2s − 1, n}. A collection of points in P n is a generic collection if every linear form in the defining ideal of each point has algebraically independent coefficients over Q. Similarly, we say a collection of lines is a generic collection if every linear form in the defining ideal of each line has algebraically independent coefficients over Q.
We can interpret this definition as saying generic collections are sufficiently random since a generic collection forms a dense subset of a large parameter space. Furthermore, as one should suspect, a generic collection of lines (points) is in general linear position since collections of lines (points) in general linear position are characterized by the nonvanishing of certain determinants in the coefficients of the defining linear forms and this implication is strict. The previous example demonstrates this fact, and the argument for this example can be found in Example 2.3.5.
For an example of a collection of points in general linear position, simply consider three points in P 2 such that the triple is not collinear (see Example 2.1.5). For an example of lines in general linear position, we only need to consider the collection in Example 2.1.3. This collection of lines must be in general linear position since the pair is skew to one another, meaning that the corresponding planes in V intersect trivally, and so their corresponding projective span must be a 3-dimensional projective subspace. We end this section with examples of points and lines in general linear position demonstrating Definition 2.1.4.
Example 2.1.5. Consider L 1 and L 2 in P 3 , from Example 2.1.3. These lines are in general linear position since they span P 3 . Now, let P 1 , P 2 , and P 3 be the following three points in P 2 : Each point corresponds to the span of e 1 , e 2 , and e 3 in a 3-dimensional C-vector space V , where e i are the standard basis vectors of V. Thus, any pair span a P 1 and all three span P 2 . Now, if instead we set P ′ 3 = [1 : 1 : 0], then the points P 1 , P 2 and P ′ 3 would not be in general linear position since P ′ 3 corresponds to the span of a vector which lies in the span of {e 1 , e 2 } in V ; hence, P ′ 3 is on the unique line crossing P 1 and P 2 in P 2 and the span of all three points is a P 1 .

Free Resolutions and Their Betti Numbers
A commutative Noetherian C-algebra R is said to be graded if R = i∈N R i as a C-vector space such that for all non-negative integers i and j, we have R i R j ⊆ R i+j , and is standard graded if R 0 = C and R is generated as a C-algebra by a finite set of degree 1 elements. Except when explicitly said, all rings are standard graded and Noetherian and all modules are finitely generated. Additionally, an R-module M is called graded if R is graded and M can be written as M = i∈N M i as a C-vector space such that for all non-negative integers i and j, we have Note each summand R i and M i is a C-vector space of finite dimension. Let S be the symmetric algebra of R 1 over C; i.e. S is the polynomial ring S = C[x 0 , . . . , x n ], where dim(R 1 ) = n + 1 and x 0 , . . . , x n is a C-basis of R 1 . We have a surjection S → R of standard graded C-algebras, and so R ∼ = S/J, where J is a homogeneous ideal of S and the kernel of this map.
Denote by m R the maximal homogeneous ideal of R. Note, we may view C as a graded R-module Free resolution were introduced in Hilbert (1890Hilbert ( , 1893 and the key insight was that a free resolution is a description of the structure of M. Thus, by studying the resolution of M we can study the properties and structure of M. Constructing a resolution amounts to the difficult task of repeatedly solving systems of polynomial equations and is based on the observation that if A is the matrix of the map R p → R q with respect to the fixed bases, then describing the module Ker(A) is equivalent to solving the system of R-linear equations Definition 2.2.1. The minimal graded free resolution F of an R-module M is an exact sequence of homomorphisms of finitely generated free graded R-modules such that d i−1 • d i = 0 for all i, M ∼ = F 0 /Im(d 1 ), and d i+1 (F i+1 ) ⊆ (x 0 , . . . , x n )F i for all i ≥ 0.
After choosing bases, we may represent each map in the resolution as a matrix. We can write where R(−j) denotes a rank one free module with a generator in degree j, and the numbers β R i,j (M ) are called the graded Betti numbers of M and are numerical invariants of M . The total Betti numbers of M are defined as β R i (M ) = j β R i,j (M ). When it is clear which module we are speaking about, we will write β i,j and β i to denote the graded Betti numbers and total Betti numbers, respectively. By construction, we have the equalities Finally, the free resolution F is unique up to isomorphism of complexes.
There is another sense in which a minimal resolution F is minimal. If G is any graded free resolution of an R-module M, then for some complex P, we have an isomorhpism G ∼ = F ⊕ P, where P is a direct sum of short trivial complexes of the form 0 −→ R(−p) −→ R(−p) −→ 0. ). Additionally, the condition that d i+1 (F i+1 ) ⊆ (x 0 , . . . , x n )F i ensures that no invertible elements appear in the differential matrices. Consider the following example of a minimal graded free resolution.
Example 2.2.2. Let R = C[x 0 , x 1 , x 2 , x 3 ]/(x 0 x 2 , x 0 x 3 , x 1 x 2 , x 1 x 3 ), as in Example 2.1.3. The following is the minimal graded free resolution of R over S When constructing a free resolution, there is no reason why one should expect the process to stop. Amazingly, thanks to Hilbert, we know every minimal graded free resolution over S terminates, in the sense that beyond some point within the free resolution, the free modules will be trivial.
Theorem 2.2.3. (Hilbert, 1890, 1893s Syzygy Theorem) Every finitely generated S-module has a free resolution of length at most the number of indeterminates of S.
For proof of Hilberts Syzygy Theorem, see Eisenbud (1995). Since Hilbert discovered this fact, significant progress has been made in understanding the structure and properties of finite free resolutions. Much less is known about infinite free resolutions; in fact, most resolutions are infinite.
Infinite resolutions can be difficult to study due to their intricacies and the fact that many of the techniques that worked over S do not work for infinite resolutions. Perhaps, the simplest example of an infinite free resolution can be produced from resolving C over R = C[x]/(x 2 ), which yields where each map is multiplication by x. This example will be a motivating example for our study of  (2011)). The Hilbert series of M is The Poincaré series of a module M over R is The graded Poincaré series of a module M over R is The Hilbert function and Hilbert series encode important information about an R-module, such as its dimension and multiplicity ). Interestingly, if M is an S-module, then the Hilbert series is a rational function. In fact, if we can compute the graded Betti numbers of M, then we can compute the Hilbert series, and consequently the Hilbert function since . (2.1) Once H M (t) is reduced, so that the numerator and denominator are relatively prime, the numerator is called the h-polynomial of M. The literature on Hilbert functions and the corresponding Hilbert series is vast and is still being studied, with many open questions (see Peeva and Stillman (2009)).
The Hilbert series of a coordinate ring has other uses other than just carrying numerical information. That is, it is a useful tool to prove various statements. A property that we make significant use of is the fact that the Hilbert series is additive along short exact sequences.
is a short exact sequence of finitely generated graded R-modules. Then, Furthermore, if I and J are two ideals of S such that I ⊆ J and H S/I (t) = H S/J (t), then I = J.
Determining the Hilbert function of an R-module can be incredibly difficult. However, there are certain projective varieties for which we can always compute the Hilbert function of the corresponding coordinate ring.
Theorem 2.3.3. (Carlini et al. (2012); Conca et al. (2001)) Let P be a generic collection of p points in P n and R the coordinate ring of P. The Hilbert function of R is In particular, if p ≤ n + 1, then The famous Hartshorne-Hirschowitz Theorem gives the Hilbert function in the case of a generic collection of lines.
Theorem 2.3.4. ) Let M be a generic collection of m lines in P n and R the coordinate ring of M. The Hilbert function of R is We want to note that this theorem is amazing and incredibly difficult to prove. It is so difficult that Hartshorne's and Hirschowitz's proof in the initial case of P 3 , which uses degeneration techniques by a smooth quadric surface, occupies more than half the paper. One could ask if any generalization holds for planes in P n for n ≥ 5; unfortunately, this is not known. Moreover, it is not even clear what the correct generalization should be.
We end this section with an example of a coordinate ring of a collection of lines in general linear position that is not generic.
in P 3 . These lines must be in general linear position since every pair is skew and thus spans P 3 .
The four defining ideals in S are The defining ideal for L is Resolving R = S/J over S yields the following free resolution Using Equation 2.1 yields the following Hilbert series Now, suppose that R is the coordinate ring for a generic collection of 4 lines in P 3 . By Theorem 4.4.4, R would have Hilbert series: So, L is not a generic collection of lines in P 3 .

Numerical Invariants of a Free Resolution
There are several numerical invariants of a free resolution for a module that can be used to better understand the module.
Definition 2.4.1. The projective dimension of a finitely generated R-module M is This invariant is quite interesting and measures the length of the resolution of M. A consequence of Hilbert's Syzygy Theorem is that for any finitely generated S-module M, pdim S (M ) ≤ n + 1.
The projective dimension of an R-module is interesting for another reason; that is, the projective dimension can be related to several other invariants of a module. One particular invariant is the depth of a module. The famous Auslander-Buchsbaum formula states that for any finitely generated R-module M we have so long as pdim R (M ) < ∞. Letting R = S, recovers a precise version of Hilbert's Syzygy Theorem.
In fact, this formula gives us an efficient way of computing bounds on either invariant.
Proposition 2.4.3. (Eisenbud (1995)) Let R be a Noetherian ring and suppose that is an exact sequence of finitely generated R-modules. Then Unfortunately, the rings we will study in Chapter 4 are not Cohen-Macaulay, but as we will see, some properties will still be preserved.
Another important invariant, and some could argue the most important, of the free resolution of a graded R-module M is its regularity.
Definition 2.4.4. Let M be a finitely generated graded R-module. The regularity of M is Similar to the projective dimension of a module, the regularity measures the growth of a free resolution of M. In essence, it measures the complexity of a module. Once more, Hilbert's Syzygy Theorem shows us that the regularity of every S-module is finite. A problematic feature of regularity is that, in practice, it tends to be difficult to bound. So another interesting question in commutative algebra is determining if we can obtain sharp upper bounds on the regularity of an R-module M or determine the regularity exactly.
Even though the regularity of a ring can be difficult to work with, not all hope is lost. The regularity behaves nicely over short exact sequences.
Proposition 2.4.5. (Eisenbud (1995); ) Suppose that is an exact sequence of finitely generated graded R-modules. Then To study these invariants easily, we place the graded Betti numbers of an R-module M into a

CHAPTER 3. INTRODUCTION TO KOSZUL ALGEBRAS
In this chapter, we will provide a very brief introduction to the study of Koszul algebras and their many interesting properties. This chapter is by no means a comprehensive review, and there is a fair amount of literature regarding Koszul algebras and certain stronger notions (Conca et al. (2013); Herzog et al. (2000); ).

Koszul Algebras
It was, and still is, a problem of homological algebra to compute the cohomology algebra of various augmented algebras. The canonical tool for attacking this problem used to be the bar resolution; for example, a classical result states that the cohomology algebra of a Lie algebra L may be computed using the Koszul resolution U (L) ⊗ E(L), where U (L) is the universal enveloping algebra of a Lie algebra and E(L) is the exterior algebra of L, and this resolution is a subcomplex of the bar resolution. Priddy, motivated by studying the Steenrod Algebra and the universal enveloping algebra, constructed resolutions conceptually analogous to the previous one ). Priddy called these resolutions Koszul resolutions and the algebras for which they are defined Koszul algebras.
Since their introduction by Priddy, Koszul algebras have been studied intensely by many different authors under many different names, such as homogeneous preKoszul algebras, Koszul algebras, Fröberg algebras, Priddy rings, wonderful rings, and formal rings. In fact, they have been studied so intensely that in  there are 18 different equivalent conditions for a ring to be Koszul. Their intense study is because these rings possess extraordinary homological and numerical properties and are connected with many fields of mathematics, such as representation theory, topology, and algebraic geometry. Yet, they are general enough to encompass many classes of rings throughout commutative algebra. For example, any polynomial ring, all quotients by quadratic monomial ideals, all quadratic complete intersections, the coordinate rings of Grassmannians in their Plücker embedding, and all suitably high Veronese subrings of any standard graded algebra are all Koszul (Conca et al. (2013); ).
In this chapter, we would like to collect various properties and facts related to Koszul algebras in the commutative setting and illustrate certain implications and counterexamples for various numerical properties that are neccassary for a C-algebra to be Koszul. By no means is this a complete survey of Koszul algebras in the commutative setting, but we will point the reader to sources that give a complete picture. We begin by defining a Koszul algebra. Consider the following example.
Then the minimal free resolution of C over R is Furthermore, pdim R (C) = ∞ for every v ≥ 2.
The previous example illustrates a necessary condition for an algebra to be Koszul. That is, if a C-algebra R = S/J is Koszul, then J is generated by homogeneous forms of degree 2 or 1.
Theorem 3.1.3. Let R = S/J be a Koszul algebra. Then J is generated by homogeneous forms of degree 2 or 1.
Proof. Suppose that J is minimally generated by f 1 , . . . , f m . We may assume that no f i is a linear form, since we can reduce to a nondegenerate presentation by quotienting by a regular sequence of linear forms. Consider the following map where if e i is a basis element of R(−1) n+1 , then d(e i ) =x i for i = 0, . . . , n. We aim to show that ker(d) is generated by linear elements if and only if J is quadratic. By assumption, we can write f i as the following for each i = 1, . . . , m and f ij ∈ m S . To prove the claim we will show that ker(d) is minimally generated by ij e j and the Koszul relations Observe that the relations u i and r ij clearly belong to ker(d). Let n j=0ḡ j e j ∈ ker(d).
It is immediate that This implies that belongs to the kernel of the map n j=0 Se j → (x 0 , . . . , x n ) with e j → x j for j = 0, . . . , n. Since the Koszul complex of x 0 , . . . , x n over S is acyclic this kernel is generated by the elements So, there exist polynomials p kl ∈ S such that n j=0 Thus, u i and r kl generate ker(d).
If this was not a minimal generating set, then we can omit a generator, say u 1 . Thus, there exists polynomials q i and g kl in S such that Hence, Je j .
Substituting x j into the place of e j we obtain which is a contradiction to Nakayama's lemma, since f 1 , . . . , f m form a minimal system of generators of J.
One might guess that all quadratic algebras are Koszul; unfortunately, more is needed (see Example 3.1.4). Furthermore, even showing a ring admits the Koszul property is usually difficult, but showing a ring is not Koszul is often just as difficult. One could ask if we can determine the Koszul property from computing the first few steps in the resolution of C over R. The answer to this question is no; we recall the following example due to Roos.
Example 3.1.4. (Roos (1993)) Let u be a positive natural number such that u ≥ 3. Define where I u is the ideal generated as follows For every u, the Hilbert series is H Ru (t) = 1 + 6t + 8t 2 , which is remarkably independent of u.
Moreover, the first non-linear syzygy in the resolution of C over R u is in homological position u + 1.
For u = 6, the resolution of C over R u has the following partial Betti As mentioned previously, and as Roos's family demonstrates, for an algebra to be Koszul it is not sufficient for it to be quadratic; in fact, the first counterexample of a quadratic algebra that is not Koszul was found by Lech (Example 3.2.4). Thus, a natural question to ask is which quadratic algebras are Koszul. Luckily, we only need to slightly modify Proposition 3.1.3 to recover sufficient conditions for a ring to be Koszul. We delay the proof of the statement until Section 3.3, where we can present a very elegant proof.
Theorem 3.1.5. ) Let R = S/J and J a monomial ideal generated by forms of degree 2. Then R is Koszul.

The Numerics of Koszul Algebras
The numerical properties of Koszul algebras are quite powerful when attempting to prove certain algebras are not Koszul. In this section, we compile a list of numerical properties that Koszul algebras possess and demonstrate how these properties can be used to determine if an algebra is not Koszul. Furthermore, if an algebra R possesses a numerical obstruction p to the Koszul property, then we say R is p-obstructed; otherwise, we say R is non-obstructed. We begin with a fundamental characterization of Koszul algebras.
Theorem 3.2.1. ) An algebra R is Koszul if and only if Proof. Let F be the graded minimal free resolution of C over R, where each free We aim to show that R is Koszul if and only if If R is Koszul, then the equality follows immediately. Now, suppose that R is not Koszul and that is a formal power series with (β α,α − β α ) (−1) α t α as the first non-vanishing term. Therefore, Equation 3.1 does not hold, which proves the claim.
Theorem 3.2.1 is incredibly powerful for determining when an algebra is not Koszul. Additionally, it gives, a necessary condition for a ring to be Koszul; that is, the Maclaurin series expansion of 1 H R (−t) must have non-negative coefficients. In the future, when we invert a Hilbert series and compute its Maclaurin series, we call the coefficents in the expansion β ′ i (C) the numerical Betti The downside of Theorem 3.2.1 is that if one wanted to show an algebra R is Koszul using Theorem 3.2.1, we must know the Hilbert series and the Poincaré series; which, in practice, is quite a lot to know about an algebra. Finally, an interesting feature of the previous theorem is that it shows that the Poincaré series of a Koszul algebra of C is always a rational function, which in general is not always true (see ).
We will present Lech's example, which is a quadratic algebra, which is not Koszul. Roos's example already shows that non-Koszul quadratic algebras exist, but the beauty of Roos's example is that it is a non-obstructed algebra. So far, Roos's example has no numerical obstructions to the Koszul property. Before we present Lech's example of a non-Koszul H R (t)-obstructed algebra, we recall a definition and a powerful theorem.
Definition 3.2.2. A homogeneous polynomial is said to be generic if its coefficients are algebraically independent over Q.
Theorem 3.2.3. (Hochster and Laksov (1987)) Let J be an ideal minimally generated by g generic By Theorem 3.2.3, the ideal J has the follows dimension in degree 3, Consequently, we have the following Hilbert series Inverting the Hilbert series, evaluating at −t, and calculating its Maclaurin series expansion yields a negative coefficient for the t 6 term: Thus, by Theorem 3.2.1, S/J cannot be Koszul.
A useful feature of Theorem 3.2.1, and the Hilbert series in general, is that it allows us to use theorems from analysis to analyze the behavior of H R (t) or 1 H R (t) to determine if an algebra admits the Koszul property.
Theorem 3.2.5. (Remmert, 1991, Vivanti-Pringsheim Theorem) Let the power series f (z) = ∞ v=0 a v z v have positive finite radius of convergence r and suppose that all but finitely many of its coefficients a v are real and non-negative. Then z = r is a singular point of f (z).
Remark 3.2.6. Theorems which guarantee a power series has all positive coefficients are rare; furthermore, determining when a power series has all positive coefficients is a difficult problem and is usually handled on an ad-hoc basis. Questions about the positivity of coefficients in power series of reciprocals of polynomials have applications in several fields, such as probability theory, commutative algebra, and combinatorics.
An immediate consequence of Theorem 3.2.5 is a necessary condition for a ring to be Koszul.
Proposition 3.2.7. (Reiner and Welker (2005)) If R is a Koszul algebra, then its h-polynomial h(t) has at least one real root.
has non-negative coefficients in its Maclaurin series expansion. So, by Theorem 3.2.5, if H R (t) has any real zeros, then P R C (t) will have a pole at ρ, where ρ is the radius of convergence of P R C (t). Note that h(t) has a real root exactly when h(−t) does. So, if h(t) had no real roots, then P R C (t) has no poles and would have infinite radius of convergence, a contradiction to Theorem 3.2.5.
This proposition is very useful when the h-polynomial is of low degree. For example, Lech's example of an algebra defined by 5 generic quadrics has the h-polynomial h(t) = 1 + 4t + 5t 2 , which has no real root. So, it cannot be Koszul, for two different numerical reasons.
The Koszul property not only places obstructions for the possible Hilbert series of an algebra R, but it can be rather restrictive on the Betti numbers of R, when resolved over S.
Theorem 3.2.8. (Boocher et al. (2017)) Let R = S/J be a Koszul algebra. If J is minimally generated by g elements, then the following hold: . . , g}, and if equality holds for i = 2, then J has height one and a linear resolution of length g.
. . , g}, and if equality holds for i = 2, then J is a complete intersection.
Example 3.2.9. Consider Lech's example (Example 3.2.4). The algebra R has the following (unreduced) Hilbert series So, the algebra R has the following Betti table resolved over S.
Another useful numerical condition for determining if a ring is Koszul, not special to Koszul algebras though, are the deviations of a ring. Specifically, the resolution of C over R is inductively constructed by adjoining ϵ h (R) variables at the h th iteration of the Tate construction. The numbers ϵ h (R) are called deviations of R. As mentioned, the deviations of a ring are not special to Koszul algebras and are especially useful to measure if a ring is a complete intersection. Furthermore, the deviations of a ring act as a measure for how far away a ring is from a complete intersection; if the deviations of a ring are large, then the ring is far from being a complete intersection.
Computing the deviations of a ring, in general, is difficult since we must construct a free resolution of C over R. Furthermore, they grow exponentially (Avramov (1999)). Not all hope is lost in the Koszul case though; we need to make a general observation for power series.
Observation 3.2.10. For each formal power series P (t) = 1 + i≥1 b j t j , with b j ∈ Z, there exists uniquely defined integers ϵ ′ n , such that where the product converges in the (t)-adic topology of Z [[t]].
Suppose that P 0 (t) = 1 and assume by induction that , and that ϵ ′ n is the only integer with that property.
Remark 3.2.11. The b i are defined recursively in terms of ϵ ′ n as follows In general, we have the following recursive formula where h j counts the number of times i j appears in the partition of i.
Using this observation we can compute numerically what the deviations would be if R is Koszul.
is a complete intersection; in this case ϵ h = 0 for h ≥ 3. (Avramov (1998) We have already determined that R cannot be Koszul because of the negative coefficent appearing for the t 6 term in the Maclaurin series of 1 H R (−t) . In addition, R is ϵ ′ h -obstructed. By Remark 3.2.11, we have the following numerical deviations Thus, R is not Koszul (since R is not a complete intersection).
These necessary numerical conditions give rise to many interesting questions for an algebra R; that is, which numerical conditions imply each other? For example, if an algebra R is H R (t)obstructed, then can it be ϵ ′ h -obstructed? If an algebra R is not ϵ ′ h -obstructed, then can it be H R (t)-obstructed? We give many counterexamples to these statements in the following example.
Example 3.2.13. In the following we present examples of quadratic algebras, none of which are Koszul, to demonstrate that many of necessary numerical conditions for Koszul algebras can fail while others do not. For each of the following statements let R be a C-algebra, which is not a complete intersection.
Proof. By Remark 3.2.11, we have and ϵ ′ h are all positive.
Claim 3.2.15. There exists a C-algebra satisfying (b) but not (d) Proof. See Example 3.2.9.
Claim 3.2.16. There exists a C-algebra satisfying (d) but not (b) The C-algebra S/J has the following Betti table over S. Proof Thus, showing that R satisfies (a). By Remark 3.2.11, the numerical deviations of R are  315 225 84  13 Therefore, R fails (b) and (d).
Before we prove the next claim we make use of two theorems and a definition.
Theorem 3.2.18. (Stanley (1978)) Let S = [x 0 , . . . , x n ] and J is an ideal generated by n + 2 Some explanation is needed before we present the following definition and theorem. As mentioned in Remark 3.2.6, theorems which guarantee a power series has all positive coefficients are very rare, and proving a power series has all positive coefficients is usually a difficult task. For example, a conjecture of Lewy and Friedrich, which arose from work on difference approximations to the wave equation, asks if the rational function has all positive coefficients in its Maclaurin series expansion. Szegö settled this conjecture using involved arguments on Bessel functions (Szegö (1933)). This motivated a series of papers using a range of different methods (Askey (1974); Askey and Gasper (1972); Kaluza (1933); Kauers (2007); Kauers and Zeilberger (2008); Pólya (1950)). In fact, the problem of determining whether a power series has all positive coefficients is so difficult that the question of the positivity of the coefficients in the Maclaurin series of has been wide open since 1972.
The following theorem is very interesting because it gives an algorithm to determine if the Maclaurin series of the reciprocal of an alternating polynomial in a single indeterminate has all positive coefficients. Before we present the theorem, we need a definition from graph theory.
Definition 3.2.19. Let Γ be a finite gimple graph with vertices V and edges E, in which each vertex is assigned a positive integer j, called the weight of the vertex i. Let c i,j be the number of complete subgraphs of Γ with i vertices whose weights sum to j. The vertex weighted clique polynomial of Γ is Theorem 3.2.20. (Bubenik and Gold (2011)) The associative noncommutative graded algebra is a two-sided ideal.
Claim 3.2.21. There exists an C-algebra R failing (a) but satisfying (b), (c), and (d). Proof x 5 ] and J = (q 1 , . . . , q 7 ), where q i is a generic quadrics. To prove the claim we make use of the following two results. By Theorem 3.2.18, we have the following Hilbert series where the index j on the vertex v i,j is the weight of the vertex and i counts the number of vertices.
We claim the vertex weighted clique polynomial of Γ is The graph H is a complete graph on 7 vertices with one vertex weighted 2, and every other vertex weighted 1. Through a basic count we have the following values c 5,5 = 6 c 5,6 = 15 c 6,6 = 1 c 6,7 = 1 Thus, the weighted vertex clique polynomial of H is We would like to remove the terms with degree larger than 3. Adding 20 vertices, 14 with weight 5 and 6 vertices with weight 7 removes the t 5 term and the t 7 term from the above polynomial.
Adding 14 edges between our weight 5 vertices with a single weight 1 vertex removes the t 6 term without introducing additional complete graphs larger than edges. Adding a single edge between a weight 1 vertex and a weight 7 vertex removes the t 8 term. Thus, So, by Theorem 3.2.20, the Maclaurin series expansion of has all positive coefficients.
By Remark 3.2.11, Hence, R satisfies (c) (since R is not a complete intersection).
Below is a diagram summarizing the lack of implications, and one implication.

Koszul Filtrations
Because of the difficulty in demonstrating rings are Koszul, we sometimes turn to a divide-andconquer strategy to prove algebras are Koszul.
(a) Every ideal I ∈ F is generated by elements of degree 1.
(b) The zero ideal (0) and the graded maximal ideal m R are in F.
(c) For every I ∈ F with I ̸ = 0, there exists J ∈ F such that J ⊆ I, I/J is cyclic, and J : I ∈ F.
This notion was inspired by the work of Herzog, Hibi, and Restuccia on strongly Koszul algebras (Herzog et al. (2000)). Conca, Trung, and Valla, showed that the existence of a Koszul filtration in R implies that every ideal in the filtration has a linear resolution over R; in particular R is Koszul, Theorem 3.3.2. (Conca et al. (2001b)) Let F be a Koszul filtration of a standard graded C-algebra R. Then the following hold: (a) For every J ∈ F the R-module, R/J is Koszul.
Proof. Suppose that F is a Koszul filtration of R and I, J ∈ F such that I/J = (ℓ). Consider the short exact sequence Taking a long exact sequence of Tor yields the following for every j ∈ Z.
By induction on the number of generators of J and the homological degree, we have the following Thus, the first and third terms of Equation 3.3 vanish for all i ̸ = j. The second claim immediately follows from the first, since m R ∈ F and R/m R ∼ = C.
An interesting question to consider is: if a C-algebra is Koszul, then does it admit a Koszul filtration? Unfortunately, the answer to this question is no.
Thus, R is Artinian. Furthermore, R = S/J is Koszul, since it is a complete intersection. Now, a necessary condition for an Artinian algebra to have a filtration is that the defining ideal contains a quadric of rank 1 or 2. Indeed, every nonzero linear combination of the above quadratic forms is a quadric of rank at least three.
We can now finally provide a very elegant proof of 3.1.5.
Theorem 3.1.5. ) Let R = S/J and J an ideal generated by monomials of degree 2. Then R is Koszul.
Proof. Define F to be the set of all ideals in R generated by indeterminates and suppose that , where each f i is a quadratic monomial. For any ideal P ⊆ R generated by indeterminates and x j ∈ P, the following equality holds Thus, R is Koszul by Theorem 3.3.2.

G-Quadartic and LG-quadratic Algebras
Another strategy to prove an algebra is Koszul is to prove something stronger.
Definition 3.4.1. We say R = S/J is G-quadratic if the defining ideal J has a Gröbner basis of quadrics with respect to some coordinate system of S 1 and some term order τ on S. An algebra R is LG-quadratic if there exists a G-quadratic algebra A and a regular sequence of linear forms ℓ 1 , . . . , ℓ c such that R ∼ = A/(ℓ 1 , . . . , ℓ c ).
Remark 3.4.2. It is immediate that every G-quadratic algebra is LG-quadratic.
At first, similar to the Koszul property, it may seem that the G-quadratic property is rather restrictive, but it turns out that many classical algebras in their standard coordinate systems are G-quadratic, including coordinate rings of Grassmannians, Schubert varieties, Flag varieties, Hibi rings, and many more. The disadvantage of the G-quadratic property is that it can be rather difficult to show algebras are G-quadratic; nevertheless, showing an algebra is G-quadratic can be a useful tool for demonstrating an algebra is Koszul. Before we prove every G-quadratic algebra is Koszul we need a theorem and a definition.
Definition 3.4.3. Given a fixed monomial order < on S, the initial monomial of a nonzero polynomial f ∈ S is the largest monomial in < (f ) appearing in f with a nonzero coefficient. The initial ideal of an ideal I ⊆ S is the monomial ideal Theorem 3.4.4. (Peeva (2011)) Suppose I, I 1 , and I 2 are graded ideals in S such that I ⊆ I 1 and If < is any monomial order on S, then for all i, j we have Theorem 3.4.5. If R is a G-quadratic algebra, then R is Koszul Proof. Let I 1 = I 2 = m S . Applying Theorem 3.4.4 yields the following There is an interesting relationship between an algebra and a quotient of that algebra.  (c) Either A and A/ℓ are both Koszul or both not Koszul.
Proof. Suppose that R is LG-quadratic. By definition, where ℓ 1 , . . . , ℓ c is a regular sequence of linear forms. By Theorem 3.4.5, the C-algebra A is Koszul.
Thus, induction and Proposition 3.4.6 yield the result.
An interesting class of algebras which are LG-quadratic is a complete intersection of quadrics.
Proof. Suppose that R = C[x 0 , . . . , x n ]/(q 1 , . . . , q m ) is a complete intersection of quadrics. Set A = C[y 1 , . . . , y n ]/(y 2 1 + q 1 , . . . , y 2 m + q m ), and notice that A is G-quadratic, because the initial ideal of its defining ideal with respect to the lex term order y i > x j for every i and j is (y 2 1 , . . . , y 2 m ). Furthermore, y 1 , . . . , y m is an A-sequence, which proves the claim.
LG-quadratic algebras play a very important role in the study of Koszul algebras because of the strong restrictions they have concerning their Betti numbers over S. Indeed, suppose R = S/J is a G-quadratic algebra and Q is a quadratic initial ideal of J. Since, J is generated by quadrics and then β 1 (R) = β 1 (S/Q). Consequently, we have the following inequality for every i ≥ 0. Since quotients by a regular sequence of linear forms do not affect the Betti numbers of R over S, we get the following remarkable theorem.
Theorem 3.4.9. If R = S/J is LG-quadratic and J is minimally generated by g elements, then the following inequality holds Remark 3.4.10.
It is an open question if the previous theorem holds for Koszul algebras.
In general, we have the following implications of the previous properties. Example 3.4.11 (Conca et al. (2013)). Let The algebra R is Koszul since it has the following Koszul filtration: and F has the following computed colon ideals This algebra is not LG-quadratic. First, recall that the h-polynomial does not change under lifting by a regular sequence of linear forms. Hence, to prove that R is not LG-quadratic it is enough to verify that there is no algebra with quadratic monomial relations with h-polynomial In general, if I is an ideal of S, not containing linear forms, with h-poynomial then I has codimension h 1 and exactly h 1 +1 2 − h 2 quadratic generators. So, we seek an ideal with codimension 2 and 5 monomial generators chosen among the generators (x 0 , x 1 )(x 0 , . . . , x 6 ). An exhaustive search in Macaulay2 (Grayson and Stillman) verifies that no such ideal exists.
Remark 3.4.12. This is the only known commutative example of a Koszul algebra which is not LG-quadratic. There is a non-commutative example over an exterior algebra (McCullough and Mere (2022)). Thus, it would be very interesting to produce a family of commutative non-obstructed Koszul algebras which are not LG-quadratic, similar to Roos's family.
Example 3.4.13 (Conca (2014)). We now present a C-algebra that admits the LG-quadratic property but not the G-quadratic property. Let and An exhaustive search in Macaulay2 verifies that h(t) = 1 + 3t − 3t 2 is not the h-polynomial of any quadratic monomial ideal in four variables. Thus, R cannot be G-quadratic. However, R ′ is G-quadratic since the initial ideal of the defining ideal is (x 2 0 , x 2 1 , x 0 x 3 , x 2 x 3 , x 1 x 4 , x 3 x 4 ), under the rev-lex order. Furthermore, we have the following isomorphism where x 4 is a regular on R ′ . So R is LG-quadratic, but not G-quadratic.

Introduction
Let S = C[x 0 , . . . , x n ] be a polynomial ring and J a graded homogeneous ideal of S. Following Priddy's work, we say the ring R = S/J is Koszul if the minimal graded free resolution of the field C over R is linear ). Koszul rings are ubiquitous in commutative algebra. For example, any polynomial ring, all quotients by quadratic monomial ideals, all quadratic complete intersections, the coordinate rings of Grassmannians in their Plücker embedding, and all suitably high Veronese subrings of any standard graded algebra are all Koszul ).
Because of the ubiquity of Koszul rings, it is of interest to determine when we can guarantee a coordinate ring will be Koszul. In 1992, Kempf proved the following theorem Theorem 4.2.1. (Kempf (1992)) Let P be a collection of p points in P n and R the coordinate ring of P. If the points of P are in general linear position and p ≤ 2n, then R is Koszul.
In 2001, Conca, Trung, and Valla proved a similar theorem, except for a generic collection of points.
Theorem 4.2.2. (Conca et al. (2001)) Let P be a generic collection of p points in P n and R the coordinate ring of P. Then R is Koszul if and only if p ≤ 1 + n + n 2 4 .
We aim to generalize these theorems to collections of lines. In Section 4.3, we review necessary background information and results related to Koszul algebras that we use in the other sections.
In Section 4.4, we study properties of coordinate rings of collections of lines and how they differ from coordinate rings of collections of points. In particular, we show In Section 4.5, we prove (a) If m is even and m + 1 ≤ n, then R has a Koszul filtration.
(b) If m is odd and m + 2 ≤ n, then R has a Koszul filtration.
In particular, R is Koszul in these cases.
Additionally, we show the coordinate ring of a generic collection of 5 lines in P 6 is Koszul by constructing a Koszul filtration. In Section 4.6, we prove Theorem 4.6.2. Let M be a generic collection of m lines in P n and R the coordinate ring of M.
Further, there is an exceptional example of a coordinate ring that is not Koszul; if M is a collection of 3 lines in general linear position in P 4 , then the coordinate ring R is not Koszul. In Section 4.7, we exhibit a collection of lines that is not a generic collection but the lines are in general linear position, and we give two examples of coordinate rings where each define a generic collection of lines with quadratic defining ideals but for numerical reasons each coordinate ring is not Koszul.
We end the document with a table summarizing the results of which coordinates rings are Koszul, which are not Koszul, and which are unknown.

Background
Let P n denote n-dimensional projective space obtained from a C-vector space of dimension n+1.
A commutative Noetherian C-algebra R is said to be graded if R = i∈N R i as an Abelian group such that for all non-negative integers i and j we have R i R j ⊆ R i+j , and is standard graded if R 0 = C and R is generated as a C-algebra by a finite set of degree 1 elements. Additionally, an R-module M is called graded if R is graded and M can be written as M = i∈N M i as an Abelian group such that for all non-negative integers i and j we have R i M j ⊆ M i+j . Note each summand R i and M i is a C-vector space of finite dimension. We always assume our rings are standard graded.
Let S be the symmetric algebra of R 1 over C; i.e. S is the polynomial ring S = C[x 0 , . . . , x n ], where dim(R 1 ) = n + 1 and x 0 , . . . , x n is a C-basis of R 1 . We have an induced surjection S → R of standard graded C-algebras, and so R ∼ = S/J, where J is a homogenous ideal and the kernel of this map. We say that J defines R and call this ideal J the defining ideal. Denote by m R the maximal homogeneous ideal of R. Except when explicitly said, all rings are graded and Noetherian and all modules are finitely generated. We may view C as a graded R-module since C ∼ = R/m R . The minimal graded free resolution F of an R-module M is an exact sequence of homomorphisms of finitely generated free R-modules

The function Hilb
After choosing bases, we may represent each map in the resolution as a matrix. We can write which module we are speaking about, we will write β i,j and β i to denote the graded Betti numbers and total Betti numbers, respectively. By construction, we have the equalities Two more invariants of a module are its projective dimension and relative Castelnuovo-Mumford regularity. These invariants are defined for an R-module M as follows: An R-module M is Cohen-Macaulay, if depth(M ) = dim(M ). Since R is a module over itself, we say R is a Cohen-Macaulay ring if it is a Cohen-Macaulay R-module. Cohen-Macaulay rings have been studied extensively, and the definition is sufficiently general to allow a rich theory with a wealth of examples in algebraic geometry. This notion is a workhorse in commutative algebra, and provides very useful tools and reductions to study rings (Bruns and Herzog (1993)). For example, if one has a graded Cohen-Macaulay C-algebra, then one can take a quotient by generic linear forms to produce an Artinian ring. A reduction of this kind is called an Artinian reduction and provides many useful tools to work with, and almost all homological invariants of the ring are preserved (Migliore and Patnott (2011)). Unfortunately, we will not be able to use these tools or reductions as the coordinate ring of a generic collection of lines is almost never Cohen-Macaulay, whereas the coordinate ring of a generic collection of points is always Cohen-Macaulay.
The absolute Castelnuovo-Mumford regularity, or the regularity, is denoted reg S (M ) and is the regularity of M as an S-module. There is a cohomological interpretation by local duality (Eisenbud and Goto (1984)). Set  To study these invariants, we place the graded Betti numbers of a module M into a It is worth observing that since M is finitely generated by homogenous elements of positive degree, the Hilbert series of M is a rational function. A short exact sequence of modules has a property we use extensively in this paper. If we have a short exact sequence of graded S-modules Whenever we use this property, we will refer to it as the additivity property of the Hilbert series.
A standard graded C-algebra R is Koszul if C has a linear R-free resolution; that is, β R i,j (C) = 0 for i ̸ = j. Koszul algebras possess remarkable homological properties. For example (a) Every finitely generated R-module has finite regularity.
(b) The residue field has finite regularity.
Koszul rings possess other interesting properties as well. Fröberg showed in  that R is Koszul if and only if H R (t) and the P R C (t) have the following relationship In general, the Poincaré series of C as an R-module can be irrational , but if R is Koszul, then Equation (4.2) tells us the Poincaré series is always rational. So a necessary condition for a coordinate ring R to be Koszul is P R C (t) = 1 H R (−t) must have non-negative coefficients in its Maclaurin series. Another necessary condition is that if R is Koszul, then the defining ideal has a minimal generating set of forms of degree at most 2. This is easy to see since . Unfortunately, the converse does not hold, but Fröberg showed that if the defining ideal is generated by monomials of degree at most 2, then R is Koszul.
Theorem 4.3.4. ) If R = S/J and J is a monomial ideal with each monomial having degree at most 2, then R is Koszul.
More generally, if J has a Gröbner basis of quadrics in some term order, then R is Koszul. If such a basis exists, we say that R is G-quadratic. More generally, R is LG-quadratic if there is a G-quadratic ring A and a regular sequence of linear forms l 1 , . . . , l r such that R ∼ = A/(l 1 , . . . , l r ).
It is worth noting that every G-quadratic ring is LG-quadratic, and every LG-quadratic ring is Koszul and that all of these implications are strict ). We briefly discuss in Section 4.7 if coordinate rings of generic collections of lines are G-quadratic or LG-quadratic.
We now define a very useful tool in proving rings are Koszul. Conca, Trung, and Valla showed in Conca et al. (2001) that if R has a Koszul filtration, then R is Koszul. In fact, a stronger statement is true. linear forms. We can see this because a point is an intersection of n hyperplanes and a line is an intersection of n − 1 hyperplanes. Also, if K is the defining ideal for P and J is the defining ideal for M, then dim C (K 1 ) = n + 1 − p and dim C (J 1 ) = n + 1 − 2m, provided either quantity is non-zero. Remark 4.4.2. We would like to note that when n = 2, R is a hypersurface and so pdim S (R) = 1, depth(R) = 2, and dim(R) = 2. Thus, we restrict our attention to the case n ≥ 3. Furthermore, an identical proof shows that if P is a collection of points in general linear position in P n and R is the coordinate ring of P, then pdim S (R) = n, depth(R) = 1, and dim(R) = 1. Hence, R is Cohen-Macaulay.

Properties of Coordinate Rings of Lines
In Conca et al. (2001), Conca, Trung, and Valla used the Hilbert function of points in P n in general linear position to prove the corresponding coordinate ring is Koszul, provided the number of points is at most 2n + 1. There is a generalization for the Hilbert function to a generic collection of points. We present both together as a single theorem for completeness, we do not use the Hilbert function for a generic collection of points.
Theorem 4.4.3. (Carlini et al. (2012); Conca et al. (2001)) Suppose that P is a collection of p points in P n . If P is a generic collection, or P is a collection in general linear position with p ≤ 2n + 1, then the Hilbert function of R is In particular, if p ≤ n + 1, then Since we aim to generalize Theorems 4.2.1 and 4.2.2, we would like to know the Hilbert series of the coordinate ring of a generic collection of lines. The famous Hartshorne-Hirschowitz Theorem provides an answer.
Theorem 4.4.4. ) Let M be a generic collection of m lines in P n and R the coordinate ring of M. The Hilbert function of R is This theorem is very difficult to prove. One could ask if any generalization holds for planes, and unfortunately, this is not known and is an open problem. Interestingly, this theorem allows us to determine the regularity for the coordinate ring R of a generic collection of lines. Now, we claim that reg S (S/K) = β ∈ {α, α − 1}. To prove this we need two inequalities: m − 2 ≥ β and n+β β+1 ≥ n(m − 1). We have the first inequality since Thus, m − 2 ≥ β. We have the second inequality, since by assumption and rearranging terms gives n + β β + 1 ≥ n(m − 1).
These inequalities together yield the following Hence, β + 1 ≥ α. Furthermore, the inequality Consider the short exact sequence If β = α, then Theorem 4.4.4 and the additive property of the Hilbert series yields the following and similarly if β = α − 1, then is positive since α is the smallest non-negative integer such that n + α α ≥ m(α + 1).
Remark 4.4.6. By Proposition 4.4.1, the coordinate ring R for a generic collection of lines is not Cohen-Macaulay, but reg S (R) = α, where α is precisely the smallest non-negative integer where Compare the previous result with the following general regularity bound for intersections of ideals generated by linear forms. The assumption that R is a coordinate ring of a generic collection of lines tells us the regularity exactly, which is much smaller then the Derksen-Sidman bound for a fixed n. By way of comparison we compute the following estimate. Proof. Let p(x) = (x + n) · · · (x + 2) − n!m. The polynomial p(x) has a unique positive root by the Intermediate Value Theorem, since the (x + n) · · · (x + 2) is increasing on the non-negative real numbers. Let a be this positive root, and observe that the smallest non-negative integer α satisfying the inequality n+α α ≥ m(α + 1) is precisely the ceiling of the root a.
We now use an inequality of Minkowski (Frenkel and Horváth (2014)). If x k and y k are positive for each k, then Thus, n−1 √ n!m = n−1 (a + n) · · · (a + 2) Taking ceilings gives the inequality.
We would like to note that reg S (R) is roughly asymptotic to the upper bound. Proposition 4.4.1 and Theorem 4.4.5 tell us the coordinate ring R of a non-trivial generic collection of lines in P n is not Cohen-Macaulay, pdim S (R) = n, and the regularity is the smallest non-negative integer α satisfying n+α α ≥ m(α + 1). So, the resolution of R is well-behaved, in the sense that if n is fixed and we allow m to vary we may expect the regularity to be low compared to the number of lines in our collection.

Koszul Filtrations for Collections of Lines
In this section we determine when a generic collection of lines, or a collection of lines in general linear position, will yield a Koszul coordinate ring. To this end, most of the work will be in constructing a Koszul filtration in the coordinate ring of a generic collection of lines. and R the coordinate ring of M. If n + 1 ≥ 2m, then after a change of basis the defining ideal is minimally generated by monomials of degree at most 2. Thus, R is Koszul.
Proof. We use· to denote a term removed from a sequence. Let R be the coordinate ring of M with defining ideal J. Through a change of basis and Remark 4.3.8 we may assume the defining ideal for each line has the following form L i = (x 0 , . . . ,x n−2i+1 ,x n−2i+2 , . . . , x n−1 , x n ), for i = 1, . . . , m. Since every L i is monomial, so is J. Furthermore, since n + 1 ≥ 2m, the reg S (R) ≤ 1. Thus, J is generated by monomials of degree at most 2. Theorem 4.3.4 guarantees R is Koszul.
Unfortunately, the simplicity of the previous proof does not carry over for larger generic collections of lines. We need a lemma.
Lemma 4.5.2. Let M be a generic collection of m lines in P n and R the coordinate ring of M. If reg S (R) = 1, then the Hilbert series of R is If reg S (R) = 2, then the Hilbert series of R is Proof. By Theorem 4.4.5, the regularity is the smallest non-negative integer α satisfying n+α α ≥ m(α + 1). Suppose reg S (R) = 1. By Theorem 4.4.4, the Hilbert series for R is Now, suppose reg S (R) = 2. By Theorem 4.4.4, the Hilbert series for R is H R (t) = 1 + (n + 1)t + 3mt 2 + 4mt 3 + · · · We can now construct a Koszul filtration for the coordinate ring of certain larger generic collections of lines.
Theorem 4.5.3. Let M be a generic collection of m lines in P n such that n ≥ 3 and m ≥ 3 and R the coordinate ring of M.
(a) If m is even and m + 1 ≤ n, then R has a Koszul filtration.
(b) If m is odd and m + 2 ≤ n, then R has a Koszul filtration.
In particular, R is Koszul in these cases.
We can now define a Koszul filtration F for R. We use· to denote the image of an element of S in R = S/J for the remainder of the paper. We have already seen in Equation (4.5) that Hence, n−k+1 linearly independent linear forms are in a minimal generating set of (J +(x 0 )) : (x 1 ).
Claim 4.5.4. The ideal (x 0 ,x 1 ,l 0 ,l 1 ) in R has Hilbert series H R/(x 0 ,x 1 ,l 0 ,l 1 ) (t) = 1 + (n − 3)t and any ideal P containing this ideal has the property that P : (ℓ) = m R , where ℓ is a linear form not contained in P.

Continuing in this fashion gives
So, both (x 0 ) : (x 1 ) and V n−k have the same Hilbert series. Furthermore, V n−k ⊆ (x 0 ) : (x 1 ). So these ideals are in fact equal. Interchanging x 0 and x 1 with l 0 and l 1 yields the remaining equality.
proving the claim.
Using Claim 4.5.6 and the additivity of the Hilbert series along the short exact sequence (4.10) yields Interchangingx 0 andx 1 withl 0 andl 1 , proves the other two equalities.
This completes the proof of Theorem 4.5.3.
There is at least one example of a coordinate ring with the Koszul property which is not covered by our previous theorem. Let M be a generic collection of 5 lines in P 6 . By Remark 4.3.8 and a change of basis we may assume the defining ideals for our 5 lines have the following form where a, b ∈ C are algebraically independent over Q. Some further explanation is needed why we may assume our 5 lines have this form.
Through a change of basis we may reduce the coefficient on x 5 in l 1 to 1, and then normalize l 3 to be monic in x 5 ; then through another change of basis we may reduce the coefficient on x 0 in l 3 to 1, and then normalize l 2 to be monic in x 0 ; then through another change of basis we may reduce the coefficient on x 6 in l 2 to 1, and then normalize l 0 to be monic in x 6 ; then through another change of basis we may reduce the coefficient on x 2 in l 0 to 1; then through another change of basis we may reduce the coefficient on x 4 in l 2 to 1; then through another change of basis we may reduce the coefficient on x 1 in l 3 to 1. Ultimately, we obtain Note the order in which we make these reductions is important.
Proposition 4.5.10. Let M be a generic collection of 5 lines in P 6 and R the coordinate ring.
Then R is Koszul.

Hilbert Function Obstructions to the Koszul Property
In this section, we determine when the coordinate ring of a generic collection of lines is not Koszul. But first, we need a theorem from Complex Analysis.
Theorem 4.6.1. (Remmert, 1991, Vivanti-Pringsheim Theorem) Let the power series f (z) = a v z v have positive finite radius of convergence r and suppose that all but finitely many of its coefficients a v are real and non-negative. Then z = r is a singular point of f (z).
Theorem 4.6.2. Let M be a generic collection of m lines in P n with n ≥ 2 and R the coordinate ring of M. If m > 1 72 3(n 2 + 10n + 13) + 3(n − 1) 3 (3n + 5) , then R is not Koszul.
Denote p(t) = 1 + (1 − n)t + (3m − 2n − 1)t 2 + (2m − n − 1)t 3 and note the leading coefficient is positive, since n + 1 < 2m. By the Intermediate Value Theorem p(t) has a negative zero, since p(0) = 1 and p(−3) = −27m + 12n + 16 < 0, since n + 1 < 2m and 1 < m. So, the radius of convergence r of P R C (t) is finite and all the coefficients are positive. So, by Theorem 4.6.1, r must occur as a singular point of P R C (t); meaning that p(t) must have 3 real roots and one of them must be positive. Recall that if the discriminant of a cubic polynomial with real coefficients is negative, then the polynomial has 2 non-real complex roots. Thus, the discriminant of p(t) must be non-negative. The discriminant of p(t) is ∆ = −m(108m 2 − 9m(n 2 + 10n + 13) + 4(n + 2) 3 ).
4. Suppose that α ≥ 3 and R is Koszul. By Theorem 4.4.4, the defining ideal of R contains a form of degree α in a minimal generating set, where α ≥ 3. Thus, R is not quadratic, a contradiction.
Hence, R is not Koszul.
We have at least one exceptional example of a coordinate ring of a generic collection of lines that is not Koszul that the previous theorem does not handle. where l = x 3 + x 4 . Let J be the defining ideal for R and notice that K = L 1 ∩ L 2 = (x 0 , x 1 x 2 , x 1 x 4 , x 3 x 2 , x 3 x 4 ).
Thus, J is generated by 6 linearly independent quadrics and possibly cubics. The cubic x 3 x 4 l is contained in J, but is not contained in the ideal (x 0 x 1 , x 0 x 2 , x 1 x 2 , x 1 x 4 , x 2 x 3 , x 0 l), since no term divides x 2 3 x 4 . Hence, there must be a cubic generator in a minimal generating set of J. Thus, R is not Koszul.
Remark 4.6.4. Since Remark 4.3.8 says that a generic collection of lines is in general linear position, then we may use Lemma 4.5.2 to show that the coordinate ring of a generic collection of 3 lines in P 4 has the same Hilbert series as R.
So, this is not a generic collection of lines.
Example 4.7.2. Consider the coordinate ring R for 5 generic lines in P 5 . The defining ideal J for R is minimally generated by quadrics and has the following Betti table computed via Macaulay2. The ring R is not Koszul by Theorem 4.6.2. Furthermore, it is known that if R is Koszul and the defining ideal is generated by g elements, then β i,2i ≤ g i for i ∈ {2, . . . , g} Avramov et al.
(2010). The previous inequality fails for i = 2. So, this ring is not Koszul for two numerical reasons.
Example 4.7.3. Consider the coordinate ring R for 6 generic lines in P 6 . The defining ideal J for R is minimally generated by quadrics and has the following Betti table computed via Macaulay2 The Algebra is not Koszul by Theorem 4.6.2, and does not fail the aforementioned inequality.
Coordinate rings with defining ideals minimally generated by quadrics are not rare, but the previous two examples are interesting since both fail for identical reasons, and one fails for an additional numerical reason. It would be interesting to determine sufficient reasons why certain numerical conditions fail and others do not. For example, why does β i,2i ≤ g i fail in one of the previous rings but not the other.
Furthermore, we would like to add that our theorems do not cover every coordinate ring R for every generic collection of lines in P n . For the coordinate rings we could not determine, there is a possibility these rings could be LG-quadratic or G-quadratic. In every possible case computable by Macaulay2 there exists a quadratic monomial ideal whose quotient ring gives the same Hilbert series as R. There could even be some change of basis which gives a quadratic Gröbner basis.
Further, if we wanted to construct a Koszul filtration in these coordinate rings, then Proposition 4.5.10 demonstrates that there is no reason we should expect a reasonable filtration unless there is a more efficient change of basis that went unobserved. Below is a table, without m = 1 and n = 1 and n = 2, summarizing our results: No, 4.6.2 The Koszul property for the coordinate ring of m generic lines in P n

Appendix
The following is Macaulay2 code to verify that the Koszul filtration in Proposition 4.5.10 is valid. Finally, in Chapter 4, we studied the coordinate ring R of a generic collection M of m lines in P n . We determined that if m > 1 72 3(n 2 + 10n + 13) + 3(n − 1) 3 (3n + 5) , then R is not Koszul. Furthermore, we determined that: (a) If m is even and m + 1 ≤ n, then R has a Koszul filtration.
(b) If m is odd and m + 2 ≤ n, then R has a Koszul filtration.
It was observed that a generic collection of 3 lines in P 4 was an exceptional algebra that did not admit the Koszul property and that a generic collection of 5 lines in P 6 was an exceptional algebra that did admit the Koszul property. In the future, it would be interesting to completely determine if every coordinate ring of m lines in P n such that n ≤ m ≤ 1 72 3(n 2 + 10n + 13) + 3(n − 1) 3 (3n + 5) , admits the Koszul property. Computationally, these algebras do not seem to admit the G-quadratic or LG-quadratic properties. Thus, determining if these algebras admit these properties would be an interesting future project.