Cristina Bertone | Università degli Studi di Torino (original) (raw)

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Papers by Cristina Bertone

Research paper thumbnail of The cones of Hilbert functions of squarefree modules

Abstract: In this paper, we study different generalizations of the notion of squarefreeness for i... more Abstract: 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.

Research paper thumbnail of The Euler characteristic as a polynomial in the Chern classes

Abstract: In this paper we obtain some explicit expressions for the Euler characteristic of a ran... more Abstract: 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.

Research paper thumbnail of The locus of points of the Hilbert scheme with bounded regularity

Abstract: In this paper we consider the Hilbert scheme Hilb_ {p (t)}^ n parameterizing subschemes... more Abstract: In this paper we consider the Hilbert scheme Hilb_ {p (t)}^ n parameterizing subschemes of P^ n with Hilbert polynomial $ p (t) $ and we investigate its locus containing points corresponding to schemes with regularity lower than or equal to a fixed integer r'. This locus is an open subscheme of Hilb_ {p (t)}^ n and we describe a set of defining equations.

Research paper thumbnail of A division algorithm in an affine framework for flat families covering Hilbert schemes

Research paper thumbnail of Borel open covering of Hilbert schemes

Abstract: Let p (t) be an admissible Hilbert polynomial in P^ n of degree d. It is well known tha... more Abstract: Let p (t) be an admissible Hilbert polynomial in P^ n of degree d. It is well known that the Hilbert scheme Hilb_p (t)^ n can be seen as a closed subscheme of a Grassmannian, hence, by Plucker embedding, it becomes a closed subset of a suitable projective space P^ E. Unluckily, the dimension E of this projective space is generally huge, so effective computations are practically impossible.

Research paper thumbnail of Polynomial factorization and curve decomposition algorithms

Research paper thumbnail of Algorithmes de Factorisation de Polynomes et de Décomposition de Courbes

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

Abstract: In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the semigroup ring a... more Abstract: 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.

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

Abstract: In this paper we compare a torsion free sheaf FF\ FF FF on PPN\ PP^ N PPN and the free vector... more Abstract: In this paper we compare a torsion free sheaf FF\ FF FF on PPN\ PP^ N PPN and the free vector bundle oplusi=1nOPN(bi)\ oplus_ {i= 1}^ n\ OPN (b_i) oplusi=1nOPN(bi) 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\ FF FF.

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

Abstract Let J⊂ S= K [x 0,…, xn] be a monomial strongly stable ideal. The collection Mf (J) of th... more Abstract Let J⊂ S= K [x 0,…, xn] 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 J-marked family. It can be endowed with a structure of affine scheme, called a J-marked scheme. For special ideals J, J-marked schemes provide an open cover of the Hilbert scheme View the MathML source, where p (t) is the Hilbert polynomial of S/J.

Research paper thumbnail of Positivity of Chern classes for reflexive sheaves on P N

Abstract It is well known that the Chern classes ci of a rank n vector bundle on PN, generated by... more Abstract It is well known that the Chern classes ci of a rank n vector bundle on PN, generated by global sections, are non-negative if i≤ 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 ci with i≥ 4 can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for i≤ 3 we show positivity of the ci with weaker hypothesis.

Research paper thumbnail of Modular absolute decomposition of equidimensional polynomial ideals

Abstract: In this paper, we present a modular strategy which describes key properties of the abso... more Abstract: 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.

Research paper thumbnail of Modular Las Vegas algorithms for polynomial absolute factorization

Let f (X, Y)∈ Z [X, Y] be an irreducible polynomial over Q. We give a Las Vegas absolute irreduci... more Let f (X, Y)∈ Z [X, Y] be an irreducible polynomial over Q. 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 ap 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.

Research paper thumbnail of The cones of Hilbert functions of squarefree modules

Abstract: In this paper, we study different generalizations of the notion of squarefreeness for i... more Abstract: 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.

Research paper thumbnail of The Euler characteristic as a polynomial in the Chern classes

Abstract: In this paper we obtain some explicit expressions for the Euler characteristic of a ran... more Abstract: 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.

Research paper thumbnail of The locus of points of the Hilbert scheme with bounded regularity

Abstract: In this paper we consider the Hilbert scheme Hilb_ {p (t)}^ n parameterizing subschemes... more Abstract: In this paper we consider the Hilbert scheme Hilb_ {p (t)}^ n parameterizing subschemes of P^ n with Hilbert polynomial $ p (t) $ and we investigate its locus containing points corresponding to schemes with regularity lower than or equal to a fixed integer r'. This locus is an open subscheme of Hilb_ {p (t)}^ n and we describe a set of defining equations.

Research paper thumbnail of A division algorithm in an affine framework for flat families covering Hilbert schemes

Research paper thumbnail of Borel open covering of Hilbert schemes

Abstract: Let p (t) be an admissible Hilbert polynomial in P^ n of degree d. It is well known tha... more Abstract: Let p (t) be an admissible Hilbert polynomial in P^ n of degree d. It is well known that the Hilbert scheme Hilb_p (t)^ n can be seen as a closed subscheme of a Grassmannian, hence, by Plucker embedding, it becomes a closed subset of a suitable projective space P^ E. Unluckily, the dimension E of this projective space is generally huge, so effective computations are practically impossible.

Research paper thumbnail of Polynomial factorization and curve decomposition algorithms

Research paper thumbnail of Algorithmes de Factorisation de Polynomes et de Décomposition de Courbes

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

Abstract: In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the semigroup ring a... more Abstract: 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.

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

Abstract: In this paper we compare a torsion free sheaf FF\ FF FF on PPN\ PP^ N PPN and the free vector... more Abstract: In this paper we compare a torsion free sheaf FF\ FF FF on PPN\ PP^ N PPN and the free vector bundle oplusi=1nOPN(bi)\ oplus_ {i= 1}^ n\ OPN (b_i) oplusi=1nOPN(bi) 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\ FF FF.

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

Abstract Let J⊂ S= K [x 0,…, xn] be a monomial strongly stable ideal. The collection Mf (J) of th... more Abstract Let J⊂ S= K [x 0,…, xn] 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 J-marked family. It can be endowed with a structure of affine scheme, called a J-marked scheme. For special ideals J, J-marked schemes provide an open cover of the Hilbert scheme View the MathML source, where p (t) is the Hilbert polynomial of S/J.

Research paper thumbnail of Positivity of Chern classes for reflexive sheaves on P N

Abstract It is well known that the Chern classes ci of a rank n vector bundle on PN, generated by... more Abstract It is well known that the Chern classes ci of a rank n vector bundle on PN, generated by global sections, are non-negative if i≤ 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 ci with i≥ 4 can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for i≤ 3 we show positivity of the ci with weaker hypothesis.

Research paper thumbnail of Modular absolute decomposition of equidimensional polynomial ideals

Abstract: In this paper, we present a modular strategy which describes key properties of the abso... more Abstract: 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.

Research paper thumbnail of Modular Las Vegas algorithms for polynomial absolute factorization

Let f (X, Y)∈ Z [X, Y] be an irreducible polynomial over Q. We give a Las Vegas absolute irreduci... more Let f (X, Y)∈ Z [X, Y] be an irreducible polynomial over Q. 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 ap 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.