Selecta Mathematica

, 25:9

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

Article

## 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

## 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.

## References

1. 1.
Aladpoosh, T., Ballico, E.: Postulation of disjoint unions of lines and a multiple point. Rend. Semin. Mat. Univ. Politec. Torino 72(3–4), 127–145 (2014)
2. 2.
Alexander, J., Hirschowitz, A.: Polynomial interpolation in several variables. J. Algebr. Geom. 4(2), 201–222 (1995)
3. 3.
Ballico, E.: On the Hilbert functions of disjoint unions of a linear space and many lines in $${\mathbb{P}}^n$$. Int. Math. Forum 5(16), 787–798 (2010)
4. 4.
Ballico, E.: Postulation of disjoint unions of lines and a few planes. J. Pure Appl. Algebra 215(4), 597–608 (2011)
5. 5.
Ballico, E.: Postulation of disjoint unions of lines and a multiple point II. Mediterr. J. Math. 13(4), 1449–1463 (2016)
6. 6.
Ballico, E.: On the maximal rank of a general union of a multiple linear space and a generic rational curve. Bol. Soc. Mat. Mex. 22(1), 13–31 (2016)
7. 7.
Carlini, E., Catalisano, M.V., Geramita, A.V.: Bipolynomial Hilbert functions. J. Algebra 324(4), 758–781 (2010)
8. 8.
Carlini, E., Catalisano, M.V., Geramita, A.V.: 3-Dimensional sundials. Cent. Eur. J. Math. 9(5), 949–971 (2011)
9. 9.
Carlini, E., Catalisano, M.V., Geramita, A.V.: Subspace arrangements, configurations of linear spaces and the quadrics containing them. J. Algebra 362(3), 70–83 (2012)
10. 10.
Carlini, E., Catalisano, M.V., Geramita, A.V.: On the Hilbert function of lines union one non-reduced point. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XV, 69–84 (2016)
11. 11.
Ciliberto, C.: Geometric aspects of polynomial interpolation in more variables and of Waring’s problem. In: Proceedings of the European Congress of Mathematics, Barcelona (2000); in: Progress in Mathematics, pp. 289–316. Brikhaüser (2001)Google Scholar
12. 12.
Cooper, S., Harbourne, B.: Regina lectures on fat points. In: Cooper, S., Sather-Wagstaff, S. (eds.) Connections Between Algebra, Combinatorics and Geometry. Springer Proceedings in Mathematics and Statistics, vol. 76, pp. 147–187. Springer, New York (2014)
13. 13.
Derksen, H.: Hilbert series of subspace arrangements. J. Pure Appl. Algebra 209(1), 91–98 (2007)
14. 14.
Derksen, H., Sidman, J.: A sharp bound for the Castelnuovo–Mumford regularity of subspace arrangements. Adv. Math. 172(2), 151–157 (2002)
15. 15.
Dumnicki, M., Harbourne, B., Szemberg, T., Tutaj-Gasińska, H.: Linear subspaces, symbolic powers and Nagata type conjectures. Adv. Math. 252, 471–491 (2014)
16. 16.
Fatabbi, G., Harbourne, B., Lorenzini, A.: Inductively computable unions of fat linear subspaces. J. Pure Appl. Algebra 219, 5413–5425 (2015)
17. 17.
Fulton, W.: Intersection Theory. Springer, Berlin (1984)
18. 18.
Geramita, A.V., Maroscia, P., Roberts, L.G.: The Hilbert function of a reduced k-algebra. J. Lond. Math. Soc. (2) 28(3), 443–452 (1983)
19. 19.
Geramita, A.V., Orecchia, F.: On the Cohen–Macaulay type of $$s$$ lines in $$\mathbb{A}^{n+1}$$. J. Algebra 70, 116–140 (1981)
20. 20.
Harbourne, B., Roé, J.: Linear systems with multiple base points in $${\mathbb{P}}^2$$. Adv. Geom. 4(1), 41–59 (2004)
21. 21.
Hartshorne, R.: Algebraic Geometry. Springer, New York (1977)
22. 22.
Hartshorne, R., Hirschowitz, A.: Droites en position générale dans l’espace projectif. In: Aroca, J.M., Buchweitz, R., Giusti, M., Merle, M. (eds.) Algebraic Geometry. Lecture Notes in Mathematics, vol. 961, pp. 169–188. Springer, Berlin (1982)
23. 23.
Hirschowitz, A.: Sur la postulation générique des courbes rationnelles. Acta Math. 146, 209–230 (1981)