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Cristina Bertone

Publications and source records attributed to Cristina Bertone.

25 records · Page 2Linked to original sources
arXiv2012

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.

math.AC↗
arXiv2010

Modular absolute decomposition of equidimensional polynomial ideals

In this paper, we present a modular strategy which describes key properties of the absolute primary decomposition of an equidimensional polynomial ideal defined by polynomials with rational coefficients. The algorithm we design is based on the classical technique of elimination of variables and colon ideals and uses a tricky choice of prime integers to work with. Thanks to this technique, we can obtain the number of absolute irreducible components, their degree, multiplicity and also the affine Hilbert function of the reduced components (namely, their initial ideal w.r.t. a degree-compatible term ordering) .

math.AC↗
arXiv2010

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: \[ Δ(\FF):=2nc_2-(n-1)c_1^2\geq -{1/12}n^2(n^2-1)\] 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 $t$.

math.AG↗
arXiv2010

Modular Las Vegas Algorithms for Polynomial Absolute Factorization

Let $f(X,Y) \in \ZZ[X,Y]$ be an irreducible polynomial over $\QQ$. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of $f$, or more precisely, of $f$ modulo some prime integer $p$. The same idea of choosing a $p$ satisfying some prescribed properties together with $LLL$ is used to provide a new strategy for absolute factorization of $f(X,Y)$. We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to factorize some polynomials of degree up to 400.

math.AG↗
arXiv2009

Positivity of Chern Classes for Reflexive Sheaves on P^N

It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on $\PP^N$, generated by global sections, are non-negative if $i\leq n$ and vanish otherwise. This paper deals with the following question: does the above result hold for the wider class of reflexive sheaves? We show that the Chern numbers $c_i$ with $i\geq 4$ can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for $i\leq 3$ we show positivity of the $c_i$ with weaker hypothesis. We obtain lower bounds for $c_1$, $c_2$ and $c_3$ for every reflexive sheaf $\FF$ which is generated by $H^0\FF$ on some non-empty open subset and completely classify sheaves for which either of them reach the minimum allowed, or some value close to it.

math.AG↗
arXiv2008

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.

math.AG↗