Splitting type, global sections and Chern classes for torsion free sheaves on P^N

In this paper we compare a torsion free sheaf \FF on \PP^N and the free vector bundle \oplus_{i=1}^n\OPN(b_i) having same rank and splitting type. We show that the first one has always "less" global sections, while it has a higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of \FF. As a consequence we obtain a direct, easy and more general proof of the "Horrocks' splitting criterion", also holding for torsion free sheaves, and lower bounds for the Chern classes c_i(\FF(t)) of twists of \FF, only depending on some numerical invariants of \FF. Especially, we prove for rank $n$ torsion free sheaves on \PP^N, whose splitting type has no gap (i.e. $b_i\geq b_{i+1}\geq b_i-1$ for every $i=1, ...,n-1$), the following formula for the discriminant: \Delta(\FF):=2nc_2-(n-1)c_1^2\geq -{1/12}n^2(n^2-1) Finally in the case of rank $n$ reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes c_3(\FF(t)), ..., c_n(\FF(t)), for the dimension of the cohomology modules H^i\FF(t) and for the Castelnuovo-Mumford regularity of \FF; these polynomial bounds only depend only on c_1(\FF), c_2(\FF), the splitting type of \FF and $t$.Comment: Final version, 15 page

On the dimension of the minimal vertex covers semigroup ring of an unmixed bipartite graph

In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the semigroup ring associated to the set of minimal vertex covers of an unmixed bipartite graph. In this paper we relate the dimension of this semigroup ring to the rank of the Boolean lattice associated to the graph.Comment: 6 pages, Pragmatic 2008, University of Catania (Italy); corrected typo

A Borel open cover of the Hilbert scheme

Let $p(t)$ be an admissible Hilbert polynomial in \PP^n of degree $d$. The Hilbert scheme \hilb^n_p(t) can be realized as a closed subscheme of a suitable Grassmannian $\mathbb G$, hence it could be globally defined by homogeneous equations in the Plucker coordinates of $\mathbb G$ and covered by open subsets given by the non-vanishing of a Plucker coordinate, each embedded as a closed subscheme of the affine space $A^D$, $D=\dim(\mathbb G)$. However, the number $E$ of Plucker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of \hilb^n_p(t), we exhibit a new open cover, consisting of marked schemes over Borel-fixed ideals, whose number is significantly smaller than $E$. Exploiting the properties of marked schemes, we prove that these open subsets are defined by equations of degree $\leq d+2$ in their natural embedding in \Af^D. Furthermore we find new embeddings in affine spaces of far lower dimension than $D$, and characterize those that are still defined by equations of degree $\leq d+2$. The proofs are constructive and use a polynomial reduction process, similar to the one for Grobner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases.Comment: 17 pages. This version contains and extends the first part of version 2 (arXiv:0909.2184v2[math.AG]). A new extended version of the second part, with some new results, is posed at arxiv:1110.0698v3[math.AC]. The title is slightly changed. Final version accepted for publicatio

Functors of Liftings of Projective Schemes

A classical approach to investigate a closed projective scheme $W$ consists of considering a general hyperplane section of $W$, which inherits many properties of $W$. The inverse problem that consists in finding a scheme $W$ starting from a possible hyperplane section $Y$ is called a {\em lifting problem}, and every such scheme $W$ is called a {\em lifting} of $Y$. Investigations in this topic can produce methods to obtain schemes with specific properties. For example, any smooth point for $Y$ is smooth also for $W$. We characterize all the liftings of $Y$ with a given Hilbert polynomial by a parameter scheme that is obtained by gluing suitable affine open subschemes in a Hilbert scheme and is described through the functor it represents. We use constructive methods from Gr\"obner and marked bases theories. Furthermore, by classical tools we obtain an analogous result for equidimensional liftings. Examples of explicit computations are provided.Comment: 25 pages. Final version. Ancillary files available at http://wpage.unina.it/cioffifr/MaterialeCoCoALiftingGeometric

The cones of Hilbert functions of squarefree modules

In this paper, we study different generalizations of the notion of squarefreeness for ideals to the more general case of modules. We describe the cones of Hilbert functions for squarefree modules in general and those generated in degree zero. We give their extremal rays and defining inequalities. For squarefree modules generated in degree zero, we compare the defining inequalities of that cone with the classical Kruskal-Katona bound, also asymptotically.Comment: 17 pages, 2 figures. This paper was produced during Pragmatic 201

The Euler characteristic as a polynomial in the Chern classes

In this paper we obtain some explicit expressions for the Euler characteristic of a rank n coherent sheaf F on P^N and of its twists F(t) as polynomials in the Chern classes c_i(F), also giving algorithms for the computation. The employed methods use techniques of umbral calculus involving symmetric functions and Stirling numbers.Comment: 12 page

Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

Let $J\subset S=K[x_0,...,x_n]$ be a monomial strongly stable ideal. The collection \Mf(J) of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme \hilbp, where $p(t)$ is the Hilbert polynomial of $S/J$. Those ideals more suitable to this aim are the $m$-truncation ideals $\underline{J}_{\geq m}$ generated by the monomials of degree $\geq m$ in a saturated strongly stable monomial ideal $\underline{J}$. Exploiting a characterization of the ideals in \Mf(\underline{J}_{\geq m}) in terms of a Buchberger-like criterion, we compute the equations defining the $\underline{J}_{\geq m}$-marked scheme by a new reduction relation, called {\em superminimal reduction}, and obtain an embedding of \Mf(\underline{J}_{\geq m}) in an affine space of low dimension. In this setting, explicit computations are achievable in many non-trivial cases. Moreover, for every $m$, we give a closed embedding \phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow \Mf(\underline{J}_{\geq m+1}), characterize those $\phi_m$ that are isomorphisms in terms of the monomial basis of $\underline{J}$, especially we characterize the minimum integer $m_0$ such that $\phi_m$ is an isomorphism for every $m\geq m_0$.Comment: 28 pages; this paper contains and extends the second part of the paper posed at arXiv:0909.2184v2[math.AG]; sections are now reorganized and the general presentation of the paper is improved. Final version accepted for publicatio

The scheme of liftings and applications

We study the locus of the liftings of a homogeneous ideal $H$ in a polynomial ring over any field. We prove that this locus can be endowed with a structure of scheme $\mathrm L_H$ by applying the constructive methods of Gr\"obner bases, for any given term order. Indeed, this structure does not depend on the term order, since it can be defined as the scheme representing the functor of liftings of $H$. We also provide an explicit isomorphism between the schemes corresponding to two different term orders. Our approach allows to embed $\mathrm L_H$ in a Hilbert scheme as a locally closed subscheme, and, over an infinite field, leads to find interesting topological properties, as for instance that $\mathrm L_H$ is connected and that its locus of radical liftings is open. Moreover, we show that every ideal defining an arithmetically Cohen-Macaulay scheme of codimension two has a radical lifting, giving in particular an answer to an open question posed by L. G. Roberts in 1989.Comment: the presentation of the results has been improved, new section (Section 6 of this version) concerning the torus action on the scheme of liftings, more detailed proofs in Section 7 of this version (Section 6 in the previous version), new example added (Example 8.5 of this version