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The Nef Cone of the Hilbert Scheme of Points on Rational Elliptic Surfaces and the Cone Conjecture

Part of: Surfaces and higher-dimensional varieties Cycles and subschemes Birational geometry

Published online by Cambridge University Press:  08 June 2020

John Kopper
John Kopper*
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA16802
*
e-mail: [email protected]
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Abstract

We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.

Keywords

Moduli spaces of sheavesHilbert schemes of pointselliptic surfacescone conjectureklt Calabi–Yau pairs

MSC classification

Primary: 14C05: Parametrization (Chow and Hilbert schemes)
Secondary: 14J27: Elliptic surfaces 14E30: Minimal model program (Mori theory, extremal rays) 14J50: Automorphisms of surfaces and higher-dimensional varieties
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 216 - 227
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

During the preparation of this article, the author was partially supported by NSF RTG grant DMS-1246844.

References

Arcara, D. and Bertram, A., Bridgeland-stable moduli spaces for K-trivial surfaces. J. Eur. Math. Soc. 15(2013), 1–38. https://doi.org/10.4171/JEMS/354CrossRefGoogle Scholar
Bertram, A. and Coskun, I., The birational geometry of the Hilbert scheme of points on surfaces. In: Birational geometry, rational curves and arithmetic, Simons Symposia, Springer, Cham, 2013, pp. 15–55.CrossRefGoogle Scholar
Bayer, A. and Macrì, E.. Projectivity and birational geometry of Bridgeland moduli spaces. J. Amer. Math. Soc. 27(2014), 707–752. https://doi.org/10.1090/S0894-0347-2014-00790-6CrossRefGoogle Scholar
Bolognese, B., Huizenga, J., Lin, Y., Riedl, E., Schmidt, B., Woolf, M., and Zhao, X., Nef cones of Hilbert schemes of points on surfaces. Algebr. Number Theory 10(2016), 907–930. https://doi.org/10.2140/ant.2016.10.907Google Scholar
Bridgeland, T., Stability conditions on triangulated categories. Ann. of Math. (2) 166(2007), 317–345. https://doi.org/10.4007/annals.2007.166.317CrossRefGoogle Scholar
Bridgeland, T., Stability conditions on K3 surfaces. Duke Math. J. 141(2008), 241–291. https://doi.org/10.1215/S0012-7094-08-14122-5CrossRefGoogle Scholar
Fogarty, J., Algebraic families on an algebraic surface. Amer. J. Math. 90(1968), 511–521. https://doi.org/10.2307/2373541CrossRefGoogle Scholar
Fogarty, J., Algebraic families on an algebraic surface. II. The Picard scheme of the punctual Hilbert scheme. Amer. J. Math. 95(1973), 660–687. https://doi.org/10.2307/2373734CrossRefGoogle Scholar
Grassi, A. and Morrison, D., Automorphisms and the Kähler cone of certain Calabi-Yau manifolds. Duke Math. J. 71(1993), 831–838. https://doi.org/10.1215/S0012-7094-93-07130-XCrossRefGoogle Scholar
Huizenga, J., Birational geometry of moduli spaces of sheaves and Bridgeland stability. In: Surveys on recent developments in algebraic geometry, Proc. Symposia in Pure Math., American Mathematical Society, Providence, RI, 2017, pp. 101–148.Google Scholar
Kopper, J., Stability and positivity of cycles. Ph.D. thesis, University of Illinois at Chicago, 2019.Google Scholar
Looijenga, E., Rational surfaces with an anticanonical cycle. Ann. of Math. (2) 114(1981), 267–322. https://doi.org/10.2307/1971295CrossRefGoogle Scholar
Maciocia, A., Computing the walls associated to Bridgeland stability conditions on projective surfaces. Asian J. Math. 18(2014), 263–279. https://doi.org/10.4310/AJM.2014.v18.n2.a5CrossRefGoogle Scholar
Totaro, B., Hilbert’s fourteenth problem over finite fields, and a conjecture on the cone of curves. Compos. Math. 144(2008), 1176–1198. https://doi.org/10.1112/S0010437X08003667CrossRefGoogle Scholar
Totaro, B., The cone conjecture for Calabi-Yau pairs in dimension 2. Duke Math. J. 154(2010), 241–263. https://doi.org/10.1215/00127094-2010-039CrossRefGoogle Scholar