Star configuration points and generic plane curves

Proceedings of the American Mathematical Society - Tập 139 Số 12 - Trang 4181-4192
Enrico Carlini1, Adam Van Tuyl2
1Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Turin, Italy
2Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1

Tóm tắt

Let ℓ 1 , … , ℓ l \ell _1,\ldots ,\ell _l be l l lines in P 2 \mathbb {P}^2 such that no three lines meet in a point. Let X ( l ) \mathbb {X}(l) be the set of points { ℓ i ∩ ℓ j   |   1 ≤ i > j ≤ l } ⊆ P 2 \{\ell _i \cap \ell _j ~|~ 1 \leq i > j \leq l\} \subseteq \mathbb {P}^2 . We call X ( l ) \mathbb {X}(l) a star configuration. We describe all pairs ( d , l ) (d,l) such that the generic degree d d curve in P 2 \mathbb {P}^2 contains an X ( l ) \mathbb {X}(l) . Our proof strategy uses both a theoretical and an explicit algorithmic approach. We also describe how one may extend our algorithmic approach to similar problems.

Từ khóa


Tài liệu tham khảo

Bocci, Cristiano, 2010, Comparing powers and symbolic powers of ideals, J. Algebraic Geom., 19, 399, 10.1090/S1056-3911-09-00530-X

Carlini, Enrico, 2005, Codimension one decompositions and Chow varieties, 67

Carlini, Enrico, 2008, Complete intersections on general hypersurfaces, Michigan Math. J., 57, 121, 10.1307/mmj/1220879400

Carlini, Enrico, 2010, Complete intersection points on general surfaces in ℙ³, Michigan Math. J., 59, 269, 10.1307/mmj/1281531455

CoCoATeam, CoCoA: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it.

S. Cooper, B. Harbourne, Z. Teitler, Combinatorial bounds on Hilbert functions of fat points in the plane. J. Pure Appl. Algebra 215 (2011), 2165-2179.

A.V. Geramita, B. Harbourne, J. Migliore. Personal communication.

Geramita, A. V., 2006, On the first infinitesimal neighborhood of a linear configuration of points in ℙ², J. Algebra, 298, 563, 10.1016/j.jalgebra.2006.01.035

Grothendieck, Alexander, 2005, Cohomologie locale des faisceaux coh\'{e}rents et th\'{e}or\`emes de Lefschetz locaux et globaux (SGA 2), 4

Harris, Joe, 1992, Algebraic geometry, 133, 10.1007/978-1-4757-2189-8

Lefschetz, Solomon, 1921, On certain numerical invariants of algebraic varieties with application to abelian varieties, Trans. Amer. Math. Soc., 22, 327, 10.2307/1988897

Lüroth, J., 1869, Einige Eigenschaften einer gewissen Gattung von Curven vierter Ordnung, Math. Ann., 1, 37, 10.1007/BF01447385

Morley, Frank, 1919, On the Luroth Quartic Curve, Amer. J. Math., 41, 279, 10.2307/2370287

Ottaviani, Giorgio, 2010, On the hypersurface of Lüroth quartics, Michigan Math. J., 59, 365, 10.1307/mmj/1281531462

F. Severi. Una proprieta’ delle forme algebriche prive di punti multipli. Rend. Accad. Lincei, II 15 (1906) 691–696.

Szabó, Endre, 1996, Complete intersection subvarieties of general hypersurfaces, Pacific J. Math., 175, 271, 10.2140/pjm.1996.175.271

Thông tin
Thông tin xuất bản

Proceedings of the American Mathematical Society

Tập 139 Số 12

4181-4192

Thông tin tác giả