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, 25:9 | Cite as

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

  • Tahereh AladpooshEmail author
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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.

Keywords

Good postulation Specialization Degeneration Double line Double point Generic union of lines Sundial Residual scheme Hartshorne–Hirschowitz theorem Castelnuovo’s inequality 

Mathematics Subject Classification

14N05 14N20 14C17 14C20 
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Notes

Acknowledgements

I would like to thank Professor M. V. Catalisano, for sharing with me many geometrical insight about techniques involved in the postulation problem during my stay at the university of Genova, for suggesting that I study the problem considered here, and particularly for her willingness to read patiently an early version of this paper.

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Authors and Affiliations

  1. 1.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

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