Alexandra Seceleanu

Ideals with larger projective dimension and regularity

We define a family of homogeneous ideals with large projective dimension and regularity relative to the number of generators and their common degree. This family subsumes and improves upon constructions given by Caviglia (2004) and McCullough (2011). In particular, we describe a family of three-generated homogeneous ideals, in arbitrary characteristic, whose projective dimension grows asymptotically as a power of the degree of the generators. Highlights? We define a family of homogeneous ideals with large projective dimension and regularity relative to the number of generators and their common degree. ? This family subsumes and improves upon constructions given by Caviglia (2004) and McCullough (2011). ? In particular, we describe a family of three-generated homogeneous ideals, in arbitrary characteristic, whose projective dimension grows asymptotically as a power of the degree of the generators.

Resurgences for ideals of special point configurations in PN coming from hyperplane arrangements

Abstract Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [3] , [7] , [13] , [16] , [18] , [19] , [20] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [3] , [15] . There have been exciting new developments in this area recently. It had been expected for several years that I ( N r − N + 1 ) ⊆ I r should hold for the ideal I of any finite set of points in P N for all r > 0 , but in the last year various counterexamples have now been constructed (see [11] , [17] , [8] ), all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

The Waldschmidt constant for squarefree monomial ideals

Given a squarefree monomial ideal

Syzygies and singularities of tensor product surfaces of bidegree (2,1)

I \subseteq R =k[x_1,\ldots ,x_n]

Containment counterexamples for ideals of various configurations of points in PN

I⊆R=k[x1,…,xn], we show that

A homological criterion for the containment between symbolic and ordinary powers of some ideals of points in P2

\widehat{\alpha }(I)