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A bound for Castelnuovo-Mumford regularity by double point divisors
- Source :
- Advances in Mathematics. 364:107008
- Publication Year :
- 2020
- Publisher :
- Elsevier BV, 2020.
-
Abstract
- Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\text{reg} (\mathcal{O}_X)$ is not even bounded above by any polynomial function of $d$ when $X$ is not smooth. For a normality bound in the smooth case, we establish that $\text{reg}(X) \leq n(d-2)+1$, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that $\text{reg}(X) \leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\text{reg}(X)$ under suitable assumptions.<br />Comment: 23 pages. This paper has been largely rewritten after McCullough-Peeva's counterexamples to the Eisenbud-Goto regularity conjecture, which appeared in J. Amer. Math. Soc. in 2018. We also added new results on the regularity of smooth projective varieties of arbitrary dimension
- Subjects :
- Conjecture
Mathematics::Commutative Algebra
General Mathematics
010102 general mathematics
14N05, 13D02, 14N25, 51N35
Vector bundle
Codimension
Mathematics - Commutative Algebra
Commutative Algebra (math.AC)
01 natural sciences
Combinatorics
Mathematics - Algebraic Geometry
Mathematics::Algebraic Geometry
Castelnuovo–Mumford regularity
0103 physical sciences
FOS: Mathematics
010307 mathematical physics
0101 mathematics
Invariant (mathematics)
Algebraically closed field
Algebraic Geometry (math.AG)
Projective variety
Mathematics
Counterexample
Subjects
Details
- ISSN :
- 00018708
- Volume :
- 364
- Database :
- OpenAIRE
- Journal :
- Advances in Mathematics
- Accession number :
- edsair.doi.dedup.....c0dbdaef5ffdfbaa4b4eb948eb6ebb89