48 research outputs found
A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds
Fix a smooth projective 3-fold X, c1, H ∈ Pic(X) with H ample, and d ∈ Z. Assume the existence of integers a, b with a ≠ 0 such that ac1 is numerically equivalent to bH. Let M(X, 2, c1, d, H) be the moduli scheme of H-stable rank 2 vector bundles, E, on X with c1(E) = c1 and c2(E) · H = d. Let m(X, 2, c1, d, H) be the number of its irreducible components. Then lim supd→ ∞m(X, 2, c1, d, H) = +∞
Dimension of families of determinantal schemes
A scheme of codimension is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous matrix and is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers and we denote by (resp. ) the locus of good (resp. standard) determinantal schemes of codimension defined by the maximal minors of a matrix where is a homogeneous polynomial of degree .
In this paper we address the following three fundamental problems: To determine (1) the dimension of (resp. ) in terms of and , (2) whether the closure of is an irreducible component of , and (3) when is generically smooth along . Concerning question (1) we give an upper bound for the dimension of (resp. ) which works for all integers and , and we conjecture that this bound is sharp. The conjecture is proved for , and for under some restriction on and . For questions (2) and (3) we have an affirmative answer for and , and for under certain numerical assumptions
Uniform Steiner bundles
In this work we study -type uniform Steiner bundles, being the lowest degree of the splitting. We prove sharp upper and lower bounds for the rank in the case and moreover we give families of examples for every allowed possible rank and explain which relation exists between the families. After dealing with the case in general, we conjecture that every -type uniform Steiner bundle is obtained through the proposed construction technique
Togliatti systems and Galois coverings
We study the homogeneous artinian ideals of the polynomial ring generated by the homogeneous polynomials of degree d which are invariant under an action of the cyclic group , for any . We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal , where e is a primitive d-th root of the unity. We get a complete description when d is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations
On the classification of Togliatti systems
In [MeMR], Mezzetti and Mir\'{o}-Roig proved that the minimal number of generators μ(I) of a minimal (smooth) monomial Togliatti system I⊂k[x0,¿,xn] satisfies 2n+1≤μ(I)≤(n+d−1n−1) and they classify all smooth minimal monomial Togliatti systems I⊂k[x0,¿,xn] with 2n+1≤μ(I)≤2n+2. In this paper, we address the first open case. We classify all smooth monomial Togliatti systems I⊂k[x0,¿,xn] of forms of degree d≥4 with μ(I)=2n+3 and n≥2 and all monomial Togliatti systems I⊂k[x0,x1,x2] of forms of degree d≥6 with μ(I)=7
The representation type of determinantal varieties
This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves of arbitrary high rank on a general standard (resp. linear) determinantal scheme of codimension and defined by the maximal minors of a homogeneous matrix . The sheaves are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme is of wild representation type provided the degrees of the entries of the matrix satisfy some weak numerical assumptions; and (2) we determine values of and for which a linear standard determinantal scheme is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. is of Ulrich wild representation type
Klyachko diagram of monomial ideals
In this paper, we introduce the notion of a Klyachko diagram for a monomial ideal I in a certain multi-graded polynomial ring, namely the Cox ring of a smooth complete toric variety, with irrelevant maximal ideal B. We present procedures to compute the Klyachko diagram of from its monomial generators, and to retrieve the -saturation of from its Klyachko diagram. We use this description to compute the first local cohomology module . As an application, we find a formula for the Hilbert function of , and a characterization of monomial ideals with constant Hilbert polynomial, in terms of their Klyachko diagram
On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
Given any diagonal cyclic subgroup of order , let be the ideal generated by all monomials of degree which are invariants of is a monomial Togliatti system, provided , and in this case the projective toric variety parameterized by is called a -variety with group . We prove that all these -varieties are arithmetically Cohen-Macaulay and we give a combinatorial expression of their Hilbert functions. In the case , we compute explicitly the Hilbert function, polynomial and series of . 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
Irreducibility of the moduli space of orthogonal instanton bundles on Pn
In order to obtain existence criteria for orthogonal instanton bundles on , we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on and prove that every orthogonal instanton bundle with no global sections on and charge has rank . We also prove that when the rank of the bundles reaches the upper bound, , the coarse moduli space of orthogonal instanton bundles with no global sections on , with charge and rank , is affine, smooth, reduced and irreducible. Last, we construct Kronecker modules to determine the splitting type of the bundles in , whenever is non-empty
Sumsets and Veronese varieties
In this paper, to any subset we explicitly associate a unique monomial projection of a Veronese variety, whose Hilbert function coincides with the cardinality of the -fold sumsets . This link allows us to tackle the classical problem of determining the polynomial such that for all and the minimum integer for which this condition is satisfied, i.e. the so-called phase transition of . We use the Castelnuovo-Mumford regularity and the geometry of to describe the polynomial and to derive new bounds for under some technical assumptions on the convex hull of ; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties