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

Ideals with an assigned initial ideal

The stratum St(J,<) (the homogeneous stratum Sth(J,<) respectively) of a monomial ideal J in a polynomial ring R is the family of all (homogeneous) ideals of R whose initial ideal with respect to the term order < is J. St(J,<) and Sth(J,<) have a natural structure of affine schemes. Moreover they are homogeneous w.r.t. a non-standard grading called level. This property allows us to draw consequences that are interesting from both a theoretical and a computational point of view. For instance a smooth stratum is always isomorphic to an affine space (Corollary 3.6). As applications, in Sec. 5 we prove that strata and homogeneous strata w.r.t. any term ordering < of every saturated Lex-segment ideal J are smooth. For Sth(J,Lex) we also give a formula for the dimension. In the same way in Sec. 6 we consider any ideal R in k[x0,..., xn] generated by a saturated RevLex-segment ideal in k[x,y,z]. We also prove that Sth(R,RevLex) is smooth and give a formula for its dimension.Comment: 14 pages, improved version, some more example

Minimum-weight codewords of the Hermitian codes are supported on complete intersections

Let $\mathcal{H}$ be the Hermitian curve defined over a finite field $\mathbb{F}_{q^2}$. In this paper we complete the geometrical characterization of the supports of the minimum-weight codewords of the algebraic-geometry codes over $\mathcal{H}$, started in [1]: if $d$ is the distance of the code, the supports are all the sets of $d$ distinct $\mathbb{F}_{q^2}$-points on $\mathcal{H}$ complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. \texttt{DegRevLex}. For most Hermitian codes, and especially for all those with distance $d\geq q^2-q$ studied in [1], one of the two curves is always the Hermitian curve $\mathcal{H}$ itself, while if $d<q$ the supports are complete intersection of two curves none of which can be $\mathcal{H}$. Finally, for some special codes among those with intermediate distance between $q$ and $q^2-q$, both possibilities occur. We provide simple and explicit numerical criteria that allow to decide for each code what kind of supports its minimum-weight codewords have and to obtain a parametric description of the family (or the two families) of the supports. [1] C. Marcolla and M. Roggero, Hermitian codes and complete intersections, arXiv preprint arXiv:1510.03670 (2015)

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

A general framework for Noetherian well ordered polynomial reductions

Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly established. This paper presents a general definition of polynomial reduction structure, studies its features and highlights the aspects needed in order to grant and to efficiently test the main properties (noetherianity, confluence, ideal membership). The most significant aspect of this analysis is a negative reappraisal of the role of the notion of term order which is usually considered a central and crucial tool in the theory. In fact, as it was already established in the computer science context in relation with termination of algorithms, most of the properties can be obtained simply considering a well-founded ordering, while the classical requirement that it be preserved by multiplication is irrelevant. The last part of the paper shows how the polynomial basis concepts present in literature are interpreted in our language and their properties are consequences of the general results established in the first part of the paper.Comment: 36 pages. New title and substantial improvements to the presentation according to the comments of the reviewer

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