A Bound for the Castelnuovo-Mumford Regularity of Log Canonical Varieties

In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal $I$ in terms of the degrees of its generators. We assume that $I$ defines a local complete intersection with log canonical singularities.Comment: 14 pages; Title changed; Typos corrected; To appear in Journal of Pure and Applied Algebr

On a Conjecture of Cai-Zhang-Shen for Figurate Primes

A conjecture of Cai-Zhang-Shen for figurate primes says that every integer n > 1 is the sum of two figurate primes. In this paper we give respectively equivalent propositions to the conjecture in the cases of even and odd integers and then confirm the conjecture by considering functions with several variables.Comment: 12page

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