Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Recent work of Ein–Lazarsfeld–Smith and Hochster–Huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci–Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero-dimensional subschemes of projective space. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals I$I$ of smooth subschemes, generalizing what is done in Bocci and Harbourne (2010)  [5]. We apply these bounds to ideals of unions of general lines in P^N${\mathbb{P}}^N$. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in P³${\mathbb{P}}^3$.     

Every three-point set is zero dimensional

This paper answers a question of Jan J. Dijkstra by giving a proof that all three-point sets are zero dimensional. It is known that all two-point sets are zero dimensional, and it is known that for all 3$">, there are -point sets which are not zero dimensional, so this paper answers the question for the last remaining case.

     

On the Hilbert function of curvilinear zero-dimensional subschemes of projective spaces

Here we show the existence of strong restrictions for the Hilbert function of zerodimensional curvilinear subschemes of P ⁿ with one point as support and with high regularity index.     

Compactifications whose remainders are zero dimensional and retract

LetX be a locally compact non compact space. Necessary and sufficient conditions forfX/X to be a retract offX are given wherefX is the Freudenthal compactification ofX. LetX be a locally compact and zero dimensional space,m be any cardinal number andJ be a set with cardinalitym. It is proved thatX has a dyadic family of powerm if and only if there exist and compactificationY ofX such thatY/X=2 ^J andY/X is a retract ofY.     

Zero-dimensional subschemes of ruled varieties

Here we study zero-dimensional subschemes of ruled varieties, mainly Hirzebruch surfaces and rational normal scrolls, by applying the Horace method and the Terracini method This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy)     

Inflectional loci of scrolls

Let be a scroll over a smooth curve C and let denote the hyperplane bundle. The special geometry of X implies that certain sheaves related to the principal part bundles of are locally free. The inflectional loci of X can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.      

Efficient generation of zero dimensional ideals in polynomial rings

Senior Research Fellow of the Council of Scientific and Industrial Research, India.     

Sections of zero dimensional ideals over a Noetherian ring

Let A be a commutative Noetherian ring and be an ideal containing a monic polynomial such that A[T]/I is zero dimensional. Suppose the conormal module I/I ² is generated by r elements over A[T]/I. Then a set of r generators of can be lifted to a r generating set of I. A part of this work is done at the Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy. Received: 12 March 2006 Revised: 29 January 2007     

Osculating properties of decomposable scrolls

Osculating spaces of decomposable scrolls (of any genus and not necessarily normal) are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra components of the second discriminant locus – deriving from flexes – are investigated and a new class of uninflected surface scrolls is presented and characterized. Further properties related to osculation are discussed for (not necessarily decomposable) scrolls (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Smoothable zero dimensional schemes and special projections of algebraic varieties

We study the degrees of generators of the ideal of a projected Veronese variety to depending on the center of projection. This is related to the geometry of zero dimensional schemes of length 8 in , Cremona transforms of , and the geometry of Tonoli Calabi‐Yau threefolds of degree 17 in .     

Enumeration of cones over cubic scrolls

We enumerate the cones over cubic scrolls satisfying appropriate incidence conditions to lines. The first and second authors were partially supported by CNPq. The third author was partially supported by PADCT/CNPq.     

ACM vector bundles on scrolls

Here we show that certain low rank ACM vector bundles on scrolls over smooth curves are iterated extensions of line bundles. Partially supported by MIUR and GNSAGA of INDAM (Italy)     

Non-special scrolls with general moduli

Abstract  In this paper we study smooth, non-special scrolls S of degree d, genus g ≥ 0, with general moduli. In particular, we study the scheme of unisecant curves of a given degree on S. Our approach is mostly based on degeneration techniques. Keywords Ruled surfaces, Hilbert schemes of scrolls, Moduli, Embedded degenerations Mathematics Subject Classification (2000) 14J26, 14D06, 14C20, (Secondary) 14H60, 14N10