arXiv subjects
Publications and source records attributed to Hal Schenck.
We give a simple proof of the Hirzebruch-Riemann-Roch theorem for smooth complete toric varieties, based on Ishida's result that the Todd genus of a smooth complete toric variety is one.
Migliore-Miró-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,x_4] generated by powers of linear forms. This is in contrast to the analogous situation in K[x_1,x_2,x_3], where WLP always holds [H.Schenck, A.Seceleanu, Proc. A.M.S. 2010, arXiv:0911.0876]. We use the inverse system dictionary to connect I to an ideal of fat points and show that failure of WLP for powers of linear forms is connected to the geometry of the associated fat point scheme. Recent results of Sturmfels-Xu in [J. Eur. Math. Soc. 2010, arXiv:0803.0892] allow us to relate WLP to Gelfand-Tsetlin patterns. See the paper "On the weak Lefschetz property for powers of linear forms" by Migliore-Miró-Roig-Nagel [arXiv:1008.2149] for related results.
For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to A_n.
In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop. A lot of arguments presented here are scattered in the literature or constitute folklore. It was one of our aims to have a usable and easily accessible collection of examples and results.
We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and monomial ideals to obtain a formula for computing the Euler characteristic of a Weil divisor D on a complete simplicial toric variety in terms of graded pieces of the Cox ring and Stanley-Reisner ring. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of the Euler characteristic of D, to the Stanley-Reisner ideal of the fan, which is used in defining the Chow ring. The formula also follows from work of Maclagan-Smith.
Conjectures of Suciu relate the fundamental group of the complement M = C^n\A of a hyperplane arrangement A to the first resonance variety of H^*(M,Z). We describe a connection between the first resonance variety and the Orlik-Terao algebra C(A) of the arrangement. In particular, we show that non-local components of R^1(A) give rise to determinantal syzygies of C(A). As a result, Proj(C(A)) lies on a scroll, placing geometric constraints on R^1(A). The key observation is that C(A) is the homogeneous coordinate ring associated to a nef but not ample divisor on the blowup of P^2 at the singular points of A.
We study the linear syzygies of a homogeneous ideal I in a polynomial ring S, focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For a variety X and divisor D with S = Sym(H^0(D)*), what conditions on D ensure that b_(i,i+1) is nonzero? Eisenbud has shown that a decomposition D = A + B such that A and B have at least two sections gives rise to determinantal equations (and corresponding syzygies) in I_X; and conjectured that if I_2 is generated by quadrics of rank at most 4, then the last nonvanishing b_(i,i+1) is a consequence of such equations. We describe obstructions to this conjecture and prove a variant. The obstructions arise from toric specializations of the Rees algebra of Koszul cycles, and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank. This gives one answer to a question posed by Eisenbud and Koh about specializations of syzygies.
We show that an Artinian quotient of K[x, y, z] by an ideal I generated by powers of linear forms has the Weak Lefschetz property. If the syzygy bundle of I is semistable this follows from results of Brenner-Kaid; our proof works without this hypothesis, which typically does not hold.
If X is the complement of a hypersurface in P^n, then Kohno showed that the nilpotent completion of the fundamental group is isomorphic to the nilpotent completion of the holonomy Lie algebra of X. When X is the complement of a hyperplane arrangement A, the ranks phi_k of the lower central series quotients of the fundamental group of X are known for isolated examples, and for two special classes: if X is hypersolvable (in which case the quadratic closure of the cohomology ring is Koszul), or if the holonomy Lie algebra decomposes in degree three as a direct product of local components. In this paper, we use the holonomy Lie algebra to obtain a formula for phi_k when A is a subarrangement of A_n. This extends Kohno's result for braid arrangements, and provides an instance of an LCS formula for arrangements which are not decomposable or hypersolvable.
Let ${\mathcal C}= \bigcup_{i=1}^n C_i \subseteq \mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a large class of arrangements of smooth rational curves, giving an inductive tool for understanding the freeness of the module $Ω^1({\mathcal C})$ of logarithmic differential forms with pole along ${\mathcal C}$. We also show that the analog of Terao's conjecture (freeness of $Ω^1({\mathcal C})$ is combinatorially determined if ${\mathcal C}$ is a union of lines) is false in this setting.
Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle L := B_1^m_1 \otimes ... \otimes B_k^m_k. We give conditions on the m_i which guarantee that the ideal of X in P(H^0(X,L)) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.
Toric codes are evaluation codes obtained from an integral convex polytope $P \subset \R^n$ and finite field $\F_q$. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently by J. Hansen and D. Joyner. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset \R^2$ by examining Minkowski sum decompositions of subpolygons of $P$. Our results give a simple and unifying explanation of bounds of Hansen and empirical results of Joyner; they also apply to previously unknown cases.
Let A be an n by d matrix having full rank n. An orthogonal dual A^{\perp} of A is a (d-n) by d matrix of rank (d-n) such that every row of A^{\perp} is orthogonal (under the usual dot product) to every row of A. We define the orthogonal dual for arrangements by identifying an essential (central) arrangement of d hyperplanes in n-dimensional space with the n by d matrix of coefficients of the homogeneous linear forms for which the hyperplanes are kernels. If n is at least 5, we show that if the matroid (or the intersection lattice) of an n-dimensional essential arrangement A contains a modular copoint whose complement spans, then the derivation module of the orthogonally dual arrangement \A^{\perp} has projective dimension at least [n(n+2)/4] - 3,([ ] denotes ceiling).
In their paper on multiplicity bounds (1998), Herzog and Srinivasan study the relationship between the graded Betti numbers of a homogeneous ideal I in a polynomial ring R and the degree of I. For certain classes of ideals, they prove a bound on the degree in terms of the largest and smallest Betti numbers, generalizing results of Huneke and Miller (1985). The bound is conjectured to hold in general; we study this using linkage. If R/I is Cohen-Macaulay, we may reduce to the case where I defines a zero-dimensional subscheme Y. If Y is residual to a zero-scheme Z of a certain type (low degree or points in special position), then we show that the conjecture is true for I_Y.
In recent work, J. Hansen uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in the projective plane. In this paper, we generalize Hansen's results from P^2 to P^m; we also show that the hypotheses in Hansen's work may be weakened. The proof is succinct and follows by combining the Cayley-Bacharach theorem and bounds on evaluation codes obtained from reduced zero-schemes.
We study the commutative algebra of three bihomogeneous polynomials p_0,p_1,p_2 of degree (2,1) in variables x,y;z,w, assuming that they never vanish simultaneously on P^1 x P^1. Unlike the situation for P^2, the Koszul complex of the p_i is never exact. The purpose of this article is to illustrate how bigraded commutative algebra differs from the classical graded case and to indicate some of the theoretical tools needed to understand the free resolution of the ideal generated by p_0,p_1,p_2.
We study the Hilbert series of a family of ideals J_ϕgenerated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in P^{n-1}. In the three variable case this is equivalent to studying the dimension of a linear system on a blow up of P^2. The ideals that arise have the points in very special position, but because there are only seven points, we can apply results of Harbourne to obtain the classes of the negative curves. Reducing to an effective, nef divisor and using Riemann-Roch yields a formula for the Hilbert series. This proves the n=3 case of a conjecture of Postnikov and Shapiro, which they later showed true for all n. Postnikov and Shapiro observe that for a family of ideals closely related to J_ϕa similar result often seems to hold, although counterexamples exist for n=4 and n=5. Our methods allow us to prove that for n=3 an analogous formula is indeed true. We close with a counterexample to a conjecture Postnikov and Shapiro make about the minimal free resolution of these ideals.