Castelnuovo-Mumford regularity: Examples of curves and surfaces

The behaviour of Castelnuovo-Mumford regularity under ``geometric'' transformations is not well understood. In this paper we are concerned with examples which will shed some light on certain questions concerning this behaviour

Projective schemes: What is Computable in low degree?

This article first presents two examples of algorithms that extracts information on scheme out of its defining equations. We also give a review on the notion of Castelnuovo-Mumford regularity, its main properties (in particular its relation to computational issues) and different ways that were used to estimate it

A Vanishing Theorem and Asymptotic Regularity of Powers of Ideal Sheaves

Let $\mathscr{I}$ be an ideal sheaf on $P^n$. In the first part of this paper, we bound the asymptotic regularity of powers of $\mathscr{I}$ as ps-3\leq \reg \mathscr{I}^p\leq ps+e, where $e$ is a constant and $s$ is the $s$-invariant of $\mathscr{I}$. We also give the same upper bound for the asymptotic regularity of symbolic powers of $\mathscr{I}$ under some conditions. In the second part, by using multiplier ideal sheaves, we give a vanishing theorem of powers of $\mathscr{I}$ when it defines a local complete intersection subvariety with log canonical singularities.Comment: 13 pages; Corrected typos, added references, improve one of the main theorem

The reduction number and degree bound of projective subschemes

In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with almost maximal degree. We also give explicit Betti tables for almost maximal cases. Interesting examples are provided to understand our main results

Multigraded Castelnuovo-Mumford Regularity

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the associated subvariety. We also provide a criterion for testing if an ample line bundle on X gives a projectively normal embedding.Comment: 30 pages, 5 figure