Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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On the non-existence of orthogonal instanton bundles on P^{2n+1}
Get PDFIn this paper we prove that there do not exist orthogonal instanton bundles on P^{2n+1} . In order to demonstrate this fact, we propose a new way of representing the invariant, introduced by L. Costa and G. Ottaviani, related to a rank 2n instanton bundle on P^{2n+1}
Splitting types of semistable vector bundles on P^2
No full textWe show that for n ≤ 5 all generic splitting types of semistable vector bundles of rank n on P^2 on which are in principle possible by the theorem of Grauert-Mülich actually occur. We prove this by constructing examples for all possible splitting types
Derived categories of toric Fano 3-folds via the Frobenius morphism
Get PDFIn [8, Conjecture 3.6], Costa and Miró-Roig state the following conjecture:Every smooth complete toric Fano variety has a full strongly exceptional collection of line bundles. The goal of this article is to prove it for toric Fano 3-folds
On the dimension of the minimal vertex cover semigroup ring of an unmixed bipartite graph
Get PDFIn 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
Notes on symmetric and exterior depth and annihilator numbers
Get PDFWe survey and compare invariants of modules over the polynomial ring and the exterior algebra. In our considerations, we focus on the depth. The exterior analogue of depth was first introduced by Aramova, Avramov and Herzog. We state similarities between the two notion of depth and exhibit their relation in the case of squarefree modules. Work of Conca, Herzog and Hibi and Trung, respectively, shows that annihilator numbers are a meaningful generalization of depth over the polynomial ring. We introduce and study annihilator numbers over the exterior algebra. Despite some minor differences in the definition, those invariants show common behavior. In both situations a positive linear combination of the annihilator numbers can be used to bound the symmetric and exterior graded Betti numbers, respectively, from above
Dimension, depth and zero-divisors of the algebra of basic k-covers of a graph
Get PDFWe study the basic k-covers of a bipartite graph G; the algebra A(G) they span, first studied by Herzog, is the fiber cone of the Alexander dual of the edge ideal. We characterize when A(G) is a domain in terms of the combinatorics of G; it follows from a result of Hochster that when A(G) is a domain, it is also Cohen-Macaulay. We then study the dimension of A(G) by introducing a geometric invariant of bipartite graphs, the “graphical dimension”. We show that the graphical dimension of G is not larger than dim(A(G)), and equality holds in many cases (e.g. when G is a tree, or a cycle). Finally, we discuss applications of this theory to the arithmetical rank
Graded Betti numbers of ideals with linear quotient
Get PDFIn this paper we show that every ideal with linear quotients is componentwise linear. We also generalize the Eliahou-Kervaire formula for graded Betti numbers of stable ideals to homogeneous ideals with linear quotients
PRAGMATIC
Get PDFThe subsequent papers are based on results obtained during or in the sequel of the Pragmatic School 2008. The school was held from July 14 till August 1, 2009 on the campus of the University of Catania. It was devoted to combinatorial aspects of commutative algebra under the title: Free resolutions and Hilbert series: algebraic, combinatorial and geometric aspect
An algorithm to compute the Stanley depth of monomial ideals
Get PDFIn this article we describe an algorithm to compute the Stanley depth of I=J where I and J are monomial ideals. We describe also an implementation in CoCoA
Structures des Q–F–algèbres A vérifiant Ax = xAx, pour tout x in A
No full textWe give a complete characterization of Q-F-algebras A which satisfy Ax = xAx for every x in A