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Projective varieties of maximal sectional regularity · arXivDesk

1502.01769Feb 6, 2015This paper extends and generalizes some results of arXiv:1305.2355. More precisely, we do not restrict ourselves to surfaces any more. Instead we give a classification of projective varieties of maximal sectional regularity of arbitrary dimension and codimension > 2

Projective varieties of maximal sectional regularity

Markus Brodmann, Wanseok Lee, Euisung Park, Peter Schenzel

Algebraic Geometry

Abstract

We study projective varieties $X \subset \mathbb{P}^r$ of dimension $n \geq 2$ , of codimension $c \geq 3$ and of degree $d \geq c + 3$

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that are of maximal sectional regularity, i.e. varieties for which the Castelnuovo-Mumford regularity

(C)

of a general linear curve section is equal to

d -c+1

, the maximal possible value (see GruLPe). As one of the main results we classify all varieties of maximal sectional regularity. If

X

is a variety of maximal sectional regularity, then either (a) it is a divisor on a rational normal

(n+1)

-fold scroll

Y \subset \mathbb{P}^{n+3}

or else (b) there is an

n

-dimensional linear subspace

\mathbb{F} \subset \mathbb{P}^r

such that

X \cap \mathbb{F} \subset \mathbb{F}

is a hypersurface of degree

d-c+1

. Moreover, suppose that

n = 2

or the characteristic of the ground field is zero. Then in case (b) we obtain a precise description of

X

as a birational linear projection of a rational normal

d-c+1

-fold scroll.

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