Postulation of generic lines and one double line in $\PP^n$ in view of generic lines and one multiple linear space

TL;DR

This paper investigates whether adding a multiple linear space to a generic union of lines in projective space preserves good postulation, providing a complete solution for the case of lines and a double line.

Contribution

It offers a complete characterization of the postulation of lines and a double line in projective space, and proposes a conjecture for the general case involving multiple linear spaces.

Findings

Double line imposes independent conditions except in one specific case.

Identifies several exceptional schemes in the general case.

Provides partial results supporting the conjecture.

Abstract

A well-known theorem by Hartshorne--Hirschowitz (\cite{HH}) states that a generic union $\mathbb{X}\subset \PP^n$, $n\geq 3$, of lines has good postulation with respect to the linear system $|\OO_{\PP^n}(d)|$. So a question that arises naturally in studying the postulation of non-reduced positive dimensional schemes supported on linear spaces is the question whether adding a $m$-multiple linear space $m\PP^r$ to $\mathbb{X}$ can still preserve it's good postulation, which means in classical language that, whether $m\PP^r$ imposes independent conditions on the linear system $|\II_{\mathbb{X}}(d)|$. Recently, the case of $r=0$, i.e., the case of lines and one $m$-multiple point, has been completely solved by several authors (\cite{CCG4}, \cite{AB}, \cite{B1}) starting with Carlini--Catalisano--Geramita, while the case of $r>0$ was remained unsolved, and this is what we wish to investigate in this paper. Precisely, we study the postulation of a generic union of $s$ lines and one $m$-multiple linear space $m\PP^r$ in $\PP^n$, $n\geq r+2$. Our main purpose is to provide a complete answer to the question in the case of lines and one double line, which says that the double line imposes independent conditions on $|\II_{\mathbb{X}}(d)|$ except for the only case $\{n=4, s=2, d=2\}$. Moreover, we discuss an approach to the general case of lines and one $m$-multiple linear space, $(m\geq 2, r\geq 1)$, particularly, we find several exceptional such schemes, and we conjecture that these are the only exceptional ones in this family. Finally, we give some partial results in support of our conjecture.