arXiv subjects

Cristiano Bocci

Publications and source records attributed to Cristiano Bocci.

22 records · Page 2Linked to original sources

arXiv2009

Comparing powers and symbolic powers of ideals

We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.

math.AG↗

arXiv2009

The resurgence of ideals of points and the containment problem

Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our previous paper (arXiv:0706.3707) we give a complete solution in terms of the least degrees of nonzero elements of the symbolic powers, under the condition that the ideal is generated in a single degree and such that that degree is the regularity of the ideal (see Corollary 1.2). Using geometric methods, we also give complete solutions for ideals of points in the projective plane, when those points lie on an irreducible conic or when the points are general and there are at most 9 points. Finally, we give new examples of ideals all of whose symbolic powers are ordinary powers of the ideal.

math.AG↗

arXiv2005

Special effect varieties in higher dimension

Here we introduce the concept of special effect varieties in higher dimension and we generalize to the n-dimensional projective space, n>=3, the two conjectures given in AG/0410527 for the planar case. Finally, we propose some examples on the product of projective spaces and we show how these results fit with the ones of Catalisano, Geramita and Gimigliano.

math.AG↗

arXiv2004

Special effect varieties and (-1)-curves

Here we introduce the concept of special effect curve which permits to study, from a different point of view, special linear systems in P^2, i.e. linear system with general multiple base points whose effective dimension is strictly greater than the expected one. In particular we study two different kinds of special effect: the α-special effect is defined by requiring some numerical conditions, while the definition of h^1-special effect concerns cohomology groups. We state two new conjectures for the characterization of special linear systems and we prove they are equivalent to the Segre and the Harbourne-Hirschowitz ones.

math.AG↗