On codimension two subvarieties in hypersurfaces

TL;DR

This paper investigates the existence of certain codimension two subvarieties within smooth hypersurfaces in projective space, revealing new examples that challenge previous assumptions about their geometric properties.

Contribution

It demonstrates the existence of ACM codimension two subvarieties in hypersurfaces that are not simply intersections with other subvarieties, and shows cases where the normal bundle sequence does not split.

Findings

Existence of ACM codimension two subvarieties not arising as intersections.

Examples where the normal bundle sequence does not split.

Challenging previous expectations about subvariety structures.

Abstract

We show that for a smooth hypersurface $X\subset \bbP^n$ of degree at least 2, there exist arithmetically Cohen-Macaulay (ACM) codimension two subvarieties $Y\subset X$ which are not an intersection $X\cap{S}$ for a codimension two subvariety $S\subset\bbP^n$. We also show there exist $Y\subset X$ as above for which the normal bundle sequence for the inclusion $Y\subset X\subset\bbP^n$ does not split.