High Rank Linear Syzygies on Low Rank Quadrics
نویسنده
چکیده
We study the linear syzygies of a homogeneous ideal I ⊆ S = Symk(V ), focussing on the graded betti numbers bi,i+1 = dimk Tori(S/I, k)i+1. For a variety X and divisor D with V = H(D), what conditions on D ensure that bi,i+1 6= 0? In [2], Eisenbud shows 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 IX ; and in [3] conjectures that if I2 is generated by quadrics of rank ≤ 4, then the last nonvanishing bi,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 in [5].
منابع مشابه
Tropical Independence Ii: the Maximal Rank Conjecture for Quadrics
Building on our earlier results on tropical independence and shapes of divisors in tropical linear series, we give a tropical proof of the maximal rank conjecture for quadrics. We also prove a tropical analogue of Max Noether’s theorem on quadrics containing a canonically embedded curve, and state a combinatorial conjecture about tropical independence on chains of loops that implies the maximal...
متن کاملThe Eisenbud - Koh - Stillman Conjecture on Linear Syzygies
Although the relationship between minimal free resolutions and Koszul cohomology has been known for a long time, it has been difficult to find a way to fully utilize the " exterior " nature of the Koszul classes. The technique used here seems to be one way to begin to do this. We prove a conjecture of Eisenbud-Koh-Stillman on linear syzygies and in consequence a conjecture of Lazarsfeld and mys...
متن کاملSyzygies, multigraded regularity and toric varieties
Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X . Given globally generated line bundles B1, . . . , Bl on X and m1, . . . , ml ∈ N, consider the line bundle L := B m1 1 ⊗ · · · ⊗ Bl l . We give conditions on the mi which guarantee that the ideal of X in P(H (X, L)∗) is generated by quadrics and the first p syzygies are line...
متن کاملSome p-ranks Related to Hermitian Varieties
We determine the p-rank of the incidence matrix of hyperplanes of PG(n, p) and points of a nondegenerate Hermitian variety. As a corollary, we obtain new bounds for the size of caps and the existence of ovoids in finite unitary spaces. This paper is a companion to [2], in which Blokhuis and this author derive the analogous p-ranks for quadrics.
متن کاملFace Recognition Based Rank Reduction SVD Approach
Standard face recognition algorithms that use standard feature extraction techniques always suffer from image performance degradation. Recently, singular value decomposition and low-rank matrix are applied in many applications,including pattern recognition and feature extraction. The main objective of this research is to design an efficient face recognition approach by combining many tech...
متن کامل