Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1173.13021
Fall, Amadou Lamine
Bounds for the Castelnuovo-Mumford regularity of singular schemes. (Bornes pour la régularité de Castelnuovo-Mumford des schémas non lisses.) (French)
[J] Ann. Inst. Fourier 59, No. 3, 1015-1027 (2009). ISSN 0373-0956; ISSN 1777-5310/e

Let ${\Bbb P}^n$ be the projective $n$-space over a field $k$, let $X$ be a closed subscheme of ${\Bbb P}^n$ of dimension $d$ and codimension $r > 0$, and let $I_X \subset R := k[X_0,\dots ,X_n]$ be the homogeneous ideal associated to $X$. Let $D \geq 2$ be an integer and assume that $I_X$ is generated by homogeneous polynomials of degree $\leq D$. If $\text{char}\, k = 0$ and $X$ is smooth, {\it A. Bertram, L. Ein} and {\it R. Lazarsfeld} [J. Am. Math. Soc. 4, No. 3, 587--602 (1991; Zbl 0762.14012)] proved that the Castelnuovo-Mumford regularity of $I_X$ satisfies the following inequality: $$ \text{reg}(I_X) \leq r(D-1)+1. $$ {\it M. Chardin} and {\it B. Ulrich} [Am J. Math. 124, No. 6, 1103--1124 (2002; Zbl 1029.14016)] showed that this inequality remains true in the case where $X$ has isolated singularities. \par In the paper under review, the author shows that, for $k$ of arbitrary characteristic, one has: $$ \text{reg}(I_X) \leq d\, !(r(D-1)-1)+1 $$ if the dimension $\delta$ of the singular locus of $X$ is $\leq 1$, and that, for $\delta \geq 2$: $$ \text{reg}(I_X) \leq \lambda D^{(n-\delta )2^{\delta -2}} $$ where $\lambda$ is a constant depending on $n$, $d$ and $\delta$. \par The author reduces the proof of these inequalities to the case where $X$ has, except at finitely many points, locally complete intersection rational singularities using an inductive argument introduced by {\it G. Caviglia} and {\it E. Sbarra} [Compos. Math. 141, No. 6, 1365--1373 (2005; Zbl 1100.13020)]. When $X$ satisfies this additional hypothesis, the author uses the method developed by Chardin and Ulrich in the above mentioned paper.
[Iustin Coandă (Bucureşti)]

MSC 2000:: *13F20 Polynomial rings
13D02 Syzygies
14B05 Singularities (algebraic geometry)
14M06 Linkage
14F17 Vanishing theorems

Keywords: Castelnuovo-Mumford regularity; projective scheme; singular locus; locally complete intersection; rational singularity

Citations: Zbl 0762.14012; Zbl 1029.14016; Zbl 1100.13020

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]