# On complete intersections containing a linear subspace

## TL;DR

This paper studies the geometry of complete intersections containing linear subspaces, showing that for certain cases the locus is irreducible, of codimension t, and the Fano scheme is zero-dimensional with length one, implying rationality.

## Contribution

It extends known results to the case t > 0, proving irreducibility, codimension, and rationality of the locus of complete intersections containing linear subspaces.

## Key findings

- The locus W_{d,k} is irreducible and of codimension t.
- For general Y in W_{d,k}, the Fano scheme F_k(Y) is zero-dimensional with length one.
- The locus W_{d,k} is rational.

## Abstract

Consider the Fano scheme $F_k(Y)$ parameterizing $k$-dimensional linear subspaces contained in a complete intersection $Y \subset \mathbb{P}^m$ of multi-degree $\underline{d} = (d_1, \ldots, d_s)$. It is known that, if $t := \sum_{i=1}^s \binom{d_i +k}{k}-(k+1) (m-k)\leqslant 0$ and $\Pi_{i=1}^sd_i >2$, for $Y$ a general complete intersection as above, then $F_k(Y)$ has dimension $-t$. In this paper we consider the case $t> 0$. Then the locus $W_{\underline{d},k}$ of all complete intersections as above containing a $k$-dimensional linear subspace is irreducible and turns out to have codimension $t$ in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general $[Y]\in W_{\underline{d},k}$ the scheme $F_k(Y)$ is zero-dimensional of length one. This implies that $W_{\underline{d},k}$ is rational.