Postulation of generic lines and one double line in $\mathbb {P}^n$ in view of generic lines and one multiple linear space

Abstract

A well-known theorem by Hartshorne and Hirschowitz (in: Aroca, Buchweitz, Giusti, Merle (eds) Algebraic geometry. Lecture notes in mathematics, Springer, Berlin, 1982) states that a generic union $\mathbb {X}\subset \mathbb {P}^n$, $n\ge 3$, of lines has good postulation with respect to the linear system $|\mathcal {O}_{\mathbb {P}^n}(d)|$. So a question that naturally arises in studying the postulation of non-reduced positive dimensional schemes supported on linear spaces is the question whether adding a m-multiple c-dimensional linear space $m\mathbb {P}^c$ to $\mathbb {X}$ can still preserve it’s good postulation, which means in classical language that, whether $m\mathbb {P}^c$ imposes independent conditions on the linear system $|\mathcal {I}_{\mathbb {X}}(d)|$. Recently, the case of $c=0$, i.e., the case of lines and one m-multiple point, has been completely solved by several authors (Carlini et al. in Ann Sc Norm Super Pisa Cl Sci (5) XV:69–84, 2016; Aladpoosh and Ballico in Rend Semin Mat Univ Politec Torino 72(3–4):127–145, 2014; Ballico in Mediterr J Math 13(4):1449–1463, 2016) starting with Carlini–Catalisano–Geramita, while the case of $c>0$ 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 $m\mathbb {P}^c$ in $\mathbb {P}^n$, $n\ge c+2$. 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 $|\mathcal {I}_{\mathbb {X}}(d)|$ 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 c-dimensional linear space, $(m\ge 2, c\ge 1)$, 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.