The Weak Lefschetz Property and powers of linear forms in K[x,y,z]

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.Comment: 5 pages, to appear in PAM

High rank linear syzygies on low rank quadrics

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.Comment: 16 pages, 3 figure

Toric surface codes and Minkowski sums

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.Comment: 15 pages, 7 figures; This version contains some minor editorial revisions -- to appear SIAM Journal on Discrete Mathematic

Geometry of Wachspress surfaces

Let P_d be a convex polygon with d vertices. The associated Wachspress surface W_d is a fundamental object in approximation theory, defined as the image of the rational map w_d from P^2 to P^{d-1}, determined by the Wachspress barycentric coordinates for P_d. We show w_d is a regular map on a blowup X_d of P^2, and if d>4 is given by a very ample divisor on X_d, so has a smooth image W_d. We determine generators for the ideal of W_d, and prove that in graded lex order, the initial ideal of I(W_d) is given by a Stanley-Reisner ideal. As a consequence, we show that the associated surface is arithmetically Cohen-Macaulay, of Castelnuovo-Mumford regularity two, and determine all the graded betti numbers of I(W_d).Comment: 24 pages 8 figure