Subvarieties of small codimension in smooth projective varieties

TL;DR

This paper investigates the properties of subvarieties within smooth projective varieties, establishing bounds on degrees and codimensions, and classifies certain quadratic varieties based on these geometric constraints.

Contribution

It provides new bounds on degrees and codimensions of subvarieties in smooth projective varieties and classifies specific quadratic varieties with large quadrics passing through a point.

Findings

deg(X) divides deg(Y)

codim_{span(Y)}Y  codim_{\u211d^{N}}X

classification of quadratic varieties with large quadrics passing through a point

Abstract

Let X\subsetneq\mathbb{P}_{\mathbb{C}}^{N} be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m>\frac{n}{2} and X is a complete intersection or that m\geq\frac{N}{2}, we show deg(X)|deg(Y) and codim_{span(Y)}Y\geq codim_{\mathbb{P}^{N}}X, where span(Y) is the linear span of Y. These bounds are sharp. As an application, we classify smooth projective n-dimensional quadratic varieties swept out by m\geq[\frac{n}{2}]+1 dimensional quadrics passing through one point.