Rosa M. Miró-Roig

Monomial ideals, almost complete intersections and the Weak Lefschetz property

Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field, and on arithmetic properties of the exponent vectors of the monomials.

Ideals of general forms and the ubiquity of the Weak Lefschetz property

Abstract Let d 1 ,…, d r be positive integers and let I =( F 1 ,…, F r ) be an ideal generated by forms of degrees d 1 ,…, d r , respectively, in a polynomial ring R with n variables. With no further information virtually nothing can be said about I , even if we add the assumption that R / I is Artinian. Our first object of study is the case where the F i are chosen generally, subject only to the degree condition. When all the degrees are the same we give a result that says, roughly, that they have as few first syzygies as possible. In the general case, the Hilbert function of R / I has been conjectured by Froberg. In a previous work the authors showed that in many situations the minimal free resolution of R / I must have redundant terms which are not forced by Koszul (first or higher) syzygies among the F i (and hence could not be predicted from the Hilbert function), but the only examples came when r = n +1. Our second main set of results in this paper show that when n +1⩽ r ⩽2 n −2, there are again situations where there must be redundant terms. Finally, we show that if Frobergs conjecture on the Hilbert function is true then any such redundant terms in the minimal free resolution must occur in the top two possible degrees of the free module. Closely connected to the Froberg conjecture is the notion of Strong Lefschetz property, and slightly less closely connected is the Weak Lefschetz property. We also study an intermediate notion, the Maximal Rank property. We continue the description of the ubiquity of these properties, especially the Weak Lefschetz property. We show that any ideal of general forms in k [ x 1 , x 2 , x 3 , x 4 ] has the Weak Lefschetz property. Then we show that for certain choices of degrees, any complete intersection has the Weak Lefschetz property and any almost complete intersection has the Weak Lefschetz property. Finally, we show that most of the time Artinian “hypersurface sections” of zeroschemes in P 2 have the Weak Lefschetz property.

The dimension of the Hilbert scheme of Gorenstein codimension 3 subschemes

Abstract In this short note, we compute the dimension of the open subset of the Hilbert scheme, Hilbp(t)Pn, parametrizing AG closed subschemes X ⊂ Pn of codimension 3.

ON THE MINIMAL FREE RESOLUTION OF n + 1 GENERAL FORMS

O-ROIG ⁄⁄ Abstract. Let R = k(x1;:::;xn) and let I be the ideal of n + 1 generically chosen forms of degrees d1 • ¢¢¢ • dn+1. We give the precise graded Betti numbers of R=I in the following cases: † n = 3. † n = 4 and P5=1 di is even. † n = 4, P 5=1 di is odd and d2 + d3 + d4 < d1 + d5 + 4. † n is even and all generators have the same degree, a, which is even. † ( Pn+1 i=1 di) i n is even and d2 + ¢¢¢ + dn < d1 + dn+1 + n. † ( P n+1 i=1 di) i n is odd, n ‚ 6 is even, d2 + ¢¢¢ + dn < d1 + dn+1 + n and d1 + ¢¢¢ + dn i dn+1 i n ? 0. We give very good bounds on the graded Betti numbers in many other cases. We also extend a result of M. Boij by giving the graded Betti numbers for a generic compressed Gorenstein algebra (i.e. one for which the Hilbert function is maximal given n and the socle degree) when n is even and the socle degree is large. A recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands. We conjecture that if the forms all have the same degree then there are no redundant summands, and we present some evidence for this conjecture.

DIMENSION OF FAMILIES OF DETERMINANTAL SCHEMES.

A scheme X ⊂ P n+c of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous t x (t+c-1) matrix and X is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers a 0 ,a 1 ,...,a t+c-2 and b 1 ,...,b t we denote by W(b;a) C Hilb p (P n+c ) (resp. W s (b;a)) the locus of good (resp. standard) determinantal schemes X C P n+c of codimension c defined by the maximal minors of a t x (t + c - 1) matrix (f ij ) i=1,...,t j=0,...,t+c-2 where f ij ∈ k[x 0 ,x 1 ,...,x n+c ] is a homogeneous polynomial of degree a j -b i . In this paper we address the following three fundamental problems: To determine (1) the dimension of W(b;a) (resp. W s (b;a)) in terms of a j and b i , (2) whether the closure of W(b; a) is an irreducible component of Hilb p (P n+c ), and (3) when Hilb p (P n+c ) is generically smooth along W(b;a). Concerning question (1) we give an upper bound for the dimension of W(b; a) (resp. W s (b; a)) which works for all integers a 0 , a 1 ,.., a t+c-2 and b 1 ,..., b t , and we conjecture that. this bound is sharp. The conjecture is proved for 2 2, and for c > 5 under certain numerical assumptions.

On the Semistability of Instanton Sheaves Over Certain Projective Varieties

We show that instanton bundles of rank r ≤ 2n − 1, defined as the cohomology of certain linear monads, on an n-dimensional projective variety with cyclic Picard group are semistable in the sense of Mumford–Takemoto. Furthermore, we show that rank r ≤ n linear bundles with nonzero first Chern class over such varieties are stable. We also show that these bounds are sharp.

Gorenstein liaison of divisors on standard determinantal schemes and on rational normal scrolls

Abstract Let C⊂ P n be an arithmetically Cohen–Macaulay subscheme. In terms of Gorenstein liaison it is natural to ask whether C is in the Gorenstein liaison class of a complete intersection. In this paper, we study the Gorenstein liaison classes of arithmetically Cohen–Macaulay divisors on standard determinantal schemes and on rational normal scrolls. As main results, we obtain that if C is an arithmetically Cohen–Macaulay divisor on a “general” arithmetically Cohen–Macaulay surface in P 4 or on a rational normal scroll surface S⊂ P n , then C is glicci (i.e. it belongs to the Gorenstein liaison class of a complete intersection).

Derived categories of projective bundles

The goal of this short note is to prove that any projective bundle P(e) → X has a tilting bundle whose summands are line bundles whenever the same holds for X.

Geometric collections and Castelnuovo–Mumford regularity

The paper begins by overviewing the basic facts on geometric exceptional collections. Then we derive, for any coherent sheaf on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo–Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties X with a geometric collection σ. We define the notion of regularity of a coherent sheaf on X with respect to σ. We show that the basic formal properties of the Castelnuovo–Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on and for a suitable geometric collection of coherent sheaves on both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface ( n odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo–Mumford regularity of their extension by zero in .

Curves of maximum genus in range A and stick-figures

In this paper we show the existence of smooth connected space curves not contained in a surface of degree less than m, with genus maximal with respect to the degree, in half of the so-called range A. The main tool is a technique of deformation of stick-figures due to G. Floystad.