Some surfaces with non-polyhedral nef cones

Rabindranath, Ashwath

doi:10.1090/proc/13296

Some surfaces with non-polyhedral nef cones
HTML articles powered by AMS MathViewer

by Ashwath Rabindranath PDF

Proc. Amer. Math. Soc. 147 (2019), 15-20 Request permission

Abstract:

We study the nef cones of complex smooth projective surfaces and give a sufficient criterion for them to be non-polyhedral. We use this to show that the nef cone of $C \times C$, where $C$ is a complex smooth projective curve of genus at least $2$, is not polyhedral.

References

Thomas Bauer, On the cone of curves of an abelian variety, Amer. J. Math. 120 (1998), no. 5, 997–1006. MR 1646050
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
Brendan Hassett and Yuri Tschinkel, Moving and ample cones of holomorphic symplectic fourfolds, Geom. Funct. Anal. 19 (2009), no. 4, 1065–1080. MR 2570315, DOI 10.1007/s00039-009-0022-6
Thomas Bauer, Brendan Hassett and Yuri Tschinkel, Mori cones of holomorphic symplectic varieties of $k3$ type, Ann. Sci. École Norm. Sup., 2014.
Alexis Kouvidakis, Divisors on symmetric products of curves, Trans. Amer. Math. Soc. 337 (1993), no. 1, 117–128. MR 1149124, DOI 10.1090/S0002-9947-1993-1149124-5
Olivier Debarre, Lawrence Ein, Robert Lazarsfeld, and Claire Voisin, Pseudoeffective and nef classes on abelian varieties, Compos. Math. 147 (2011), no. 6, 1793–1818. MR 2862063, DOI 10.1112/S0010437X11005227
Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
Gianluca Pacienza, On the nef cone of symmetric products of a generic curve, Amer. J. Math. 125 (2003), no. 5, 1117–1135. MR 2004430
Paul Vojta, Mordell’s conjecture over function fields, Invent. Math. 98 (1989), no. 1, 115–138. MR 1010158, DOI 10.1007/BF01388847