On the shape of a pure O-sequence

An order ideal is a finite poset X of (monic) monomials such that, whenever M is in X and N divides M, then N is in X. If all, say t, maximal monomials of X have the same degree, then X is pure (of type t). A pure O-sequence is the vector, h=(1,h_1,...,h_e), counting the monomials of X in each degree. Equivalently, in the language of commutative algebra, pure O-sequences are the h-vectors of monomial Artinian level algebras. Pure O-sequences had their origin in one of Richard Stanley's early works in this area, and have since played a significant role in at least three disciplines: the study of simplicial complexes and their f-vectors, level algebras, and matroids. This monograph is intended to be the first systematic study of the theory of pure O-sequences. Our work, making an extensive use of algebraic and combinatorial techniques, includes: (i) A characterization of the first half of a pure O-sequence, which gives the exact converse to an algebraic g-theorem of Hausel; (ii) A study of (the failing of) the unimodality property; (iii) The problem of enumerating pure O-sequences, including a proof that almost all O-sequences are pure, and the asymptotic enumeration of socle degree 3 pure O-sequences of type t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) characterization; (v) A pithy connection of the ICP with Stanley's matroid h-vector conjecture; (vi) A specific study of pure O-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 in characteristic zero. As a corollary, pure O-sequences of codimension 3 and type 2 are unimodal (over any field); (vii) An analysis of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras; (viii) Some observations about pure f-vectors, an important special case of pure O-sequences.Comment: iii + 77 pages monograph, to appear as an AMS Memoir. Several, mostly minor revisions with respect to last year's versio

Reducibility of Punctual Hilbert Schemes of Cone Varieties

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ⊆ ℙN of dimension n and degree d and an integer s0 such that Hilbs(X) is reducible for all s ≥ s0. X will be a projective cone in ℙN over an arbitrary projective variety Y ⊆ ℙN-1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points. © 2013 Copyright Taylor and Francis Group, LLC

The minimal resolution conjecture for points on del Pezzo surfaces

Mustaţǎ (1997) stated a generalized version of the minimal resolution conjecture for a set Z of general points in arbitrary projective varieties and he predicted the graded Betti numbers of the minimal free resolution of IZ. In this paper, we address this conjecture and we prove that it holds for a general set Z of points on any (not necessarily normal) del Pezzo surface X ⊆ ℙ d - up to three sporadic cases - whose cardinality {pipe}Z{pipe} sits into the interval [P X(r - 1),m(r)] or [n(r),P X(r)], r ≥ 4, where P X(r) is the Hilbert polynomial of X, m(r):= dr 2 + r(2 - d) and n(r):= dr 2 + r(d - 2). As a corollary we prove: (1) Mustaţǎ's conjecture for a general set of s ≥ 19 points on any integral cubic surface in ℙ 3; and (2) the ideal generation conjecture and the Cohen-Macaulay type conjecture for a general set of cardinality s ≥ 6d + 1 on a del Pezzo surface X ⊆ ℙ d. © 2012 by Mathematical Sciences Publishers

On the arithmetic Cohen\u2013Macaulayness of varieties parameterized by Togliatti systems

Given any diagonal cyclic subgroup \u39b 82 GL (n+ 1 , k) of order d, let Id 82 k[x, \u2026 , xn] be the ideal generated by all monomials { m1, \u2026 , mr} of degree d which are invariants of \u39b. Id is a monomial Togliatti system, provided r 64(d+n-1n-1), and in this case the projective toric variety Xd parameterized by (m1, \u2026 , mr) is called a GT-variety with group \u39b. We prove that all these GT-varieties are arithmetically Cohen\u2013Macaulay and we give a combinatorial expression of their Hilbert functions. In the case n= 2 , we compute explicitly the Hilbert function, polynomial and series of Xd. We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not

Special Ulrich bundles on regular Weierstrass fibrations

The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations π: S→ P1. Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations S→ C and we deal with the (non-)existence of rank one Ulrich bundles on them

Representation type of rational ACM surfaces X ⊆ P4

The goal of this work is to determine the representation type of any smooth rational ACM surface in 4 by constructing large families of simple Ulrich bundles of arbitrary rank. It turns out that, excluding the cubic scroll, all of them are of wild representation type. In addition, we show that a general linear standard determinantal variety of codimension one or two supports indecomposable Ulrich sheaves of rank 1 and 2. © 2012 Springer Science+Business Media B.V