Resumen de Postulation of generic lines and one double line in Pn in view of generic lines and one multiple linear space
Tahereh Aladpoosh
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 X⊂Pn , n≥3 , of lines has good postulation with respect to the linear system |OPn(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 mPc to X can still preserve it’s good postulation, which means in classical language that, whether mPc imposes independent conditions on the linear system |IX(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 mPc in Pn , n≥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 |IX(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≥2,c≥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.
