Margherita Barile

Set-theoretic complete intersections on binomials

Let V be an affine toric variety of codimension r over a field of any characteristic. We completely characterize the affine toric varieties that are set-theoretic complete intersections on binomials. In particular we prove that in the characteristic zero case, V is a set-theoretic complete intersection on binomials if and only if V is a complete intersection. Moreover, if F 1 ,..., F r are binomials such that I(V) = rad(F 1 ,...,F r ), then I(V) = (F 1 ,...,F r ). While in the positive characteristic p case, V is a set-theoretic complete intersection on binomials if and only if V is completely p-glued. These results improve and complete all known results on these topics.

On the Arithmetical Rank of the Edge Ideals of Forests

We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.

ARITHMETICAL RANK OF THE CYCLIC AND BICYCLIC GRAPHS

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex, the arithmetical rank equals the projective dimension of the corresponding quotient ring.

On Ideals Whose Radical Is a Monomial Ideal

ABSTRACT We develop a new general method for constructing a polynomial ideal that has the same radical as a given monomial ideal, but has fewer generators. This provides upper bounds for the arithmetical rank of monomial ideals. An explicit formula is given for ideals generated by squarefree monomials of degree 2.

Set-theoretic complete intersections in characteristic p

We describe a class of toric varieties which are set-theoretic complete intersections only over fields of one positive characteristic p.

Arithmetical Ranks of Stanley–Reisner Ideals of Simplicial Complexes with a Cone

When a cone is added to a simplicial complex Δ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley–Reisner ideals of the original simplicial complex and the new simplicial complex Δ′. In particular, we show that the arithmetical rank of the Stanley–Reisner ideal of Δ′ equals the projective dimension of the Stanley–Reisner ring of Δ′ if the corresponding equality holds for Δ.

Arithmetical Ranks of Stanley–Reisner Ideals Via Linear Algebra

We present some examples of squarefree monomial ideals whose arithmetical rank can be computed using linear algebraic considerations.

The Stanley–Reisner Ideals of Polygons as Set-Theoretic Complete Intersections

We show that the Stanley–Reisner ideal of the one-dimensional simplicial complex whose diagram is an n-gon is always a set-theoretic complete intersection in any positive characteristic.

On empty lattice simplices in dimension 4

We give an almost complete classification of empty lattice simplices in dimension 4 using the conjectural results of Mori-Morrison-Morrison, later proved by Sankaran and Bober. In particular, all of these simplices correspond to cyclic quotient singularities, and all but finitely many of them have width bounded by 2.

On the Arithmetical Rank of an Intersection of Ideals

We determine sets of elements which, under certain conditions, generate an intersection of ideals up to radical.