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A well-known theorem by Hartshorne--Hirschowitz (\cite{HH}) states that ageneric union $\mathbb{X}\subset \PP^n$, $n\geq 3$, of lines has goodpostulation with respect to the linear system $|\OO_{\PP^n}(d)|$. So a questionthat arises naturally in studying the postulation of non-reduced positivedimensional 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 goodpostulation, which means in classical language that, whether $m\PP^r$ imposesindependent 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 beencompletely solved by several authors (\cite{CCG4}, \cite{AB}, \cite{B1})starting with Carlini--Catalisano--Geramita, while the case of $r>0$ wasremained 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 isto provide a complete answer to the question in the case of lines and onedouble 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 severalexceptional such schemes, and we conjecture that these are the only exceptionalones in this family. Finally, we give some partial results in support of ourconjecture.

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