2016/06/09 by Aladpoosh, Tahereh
#14C17 #14C20 #14N05 #14N20 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1606.02974
A well-known theorem by Hartshorne--Hirschowitz (\citeHH) states that a generic union \mathbbX⊂ \PPn, n≥ 3, of lines has good postulation with respect to the linear system |\OO\PPn(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\PPr to \mathbbX can still preserve it's good postulation, which means in classical language that, whether m\PPr imposes independent conditions on the linear system |\II_\mathbbX(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 (\citeCCG4, \citeAB, \citeB1) 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\PPr in \PPn, n≥ 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_\mathbbX(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≥ 2, r≥ 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.