Giulio Caviglia

Characteristic-free bounds for the Castelnuovo–Mumford regularity

We study bounds for the Castelnuovo–Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular, our aim is to give a positive answer to a question posed by Bayer and Mumford in What can be computed in algebraic geometry ? ( Computational algebraic geometry and commutative algebra , Symposia Mathematica, vol. XXXIV (1993), 1–48) by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyse Giustis proof, which provides the result in characteristic zero, giving some insight into the combinatorial properties needed in that context. For the general case, we provide a new argument which employs the Bayer–Stillman criterion for detecting regularity.

Bounds on the Castelnuovo-Mumford regularity of tensor products

In this paper we show how, given a complex of graded modules and knowing some partial Castelnuovo-Mumford regularities for all the modules in the complex and for all the positive homologies, it is possible to get a bound on the regularity of the zero homology. We use this to prove that if dim Tor R 1 (M, N) ≤ 1, then reg(M⊗N) < reg(M)+reg(N), generalizing results of Chandler, Conca and Herzog, and Sidman. Finally we give a description of the regularity of a module in terms of the postulation numbers of filter regular hyperplane restrictions.

Sharp upper bounds for the Betti numbers of a given Hilbert polynomial

We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.

Some Ideals with Large Projective Dimension

For an ideal I in a polynomial ring over a field, a monomial support of I is the set of monomials that appear as terms in a set of minimal generators of I. Craig Huneke asked whether the size of a monomial support is a bound for the projective dimension of the ideal. We construct an example to show that, if the number of variables and the degrees of the generators are unspecified, the projective dimension of I grows at least exponentially with the size of a monomial support. The ideal we construct is generated by monomials and binomials.

Betti tables of p-Borel-fixed ideals

In this note we provide a counterexample to a conjecture of Pardue (Thesis (Ph.D.), Brandeis University, 1994), which asserts that if a monomial ideal is p-Borel-fixed, then its

On the Lex-plus-powers Conjecture

\mathbb{N}

Generic circuits sets and general initial ideals with respect to weights

-graded Betti table, after passing to any field, does not depend on the field. More precisely, we show that, for any monomial ideal I in a polynomial ring S over the ring

The pinched Veronese is Koszul

\mathbb{Z}

Some cases of the Eisenbud-Green-Harris conjecture

of integers and for any prime number p, there is a p-Borel-fixed monomial S-ideal J such that a region of the multigraded Betti table of

On a conjecture by Kalai

J(S \otimes_{\mathbb{Z}}\ell)