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

Abstract

A well-known theorem by Hartshorne--Hirschowitz (\cite{HH}) states that a generic union $\mathbb{X}\subset \PP^n$, $n\ge 3$, 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 $\PP^n$, $n\ge r+2$. 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, $(m\ge 2,r\ge 1)$, 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.

44 pages

References in corpus (1)

• Linear subspaces, symbolic powers and Nagata type conjectures