Duke Mathematical Journal

Negative curves on algebraic surfaces

Thomas Bauer, Brian Harbourne, Andreas Leopold Knutsen, Alex Küronya, Stefan Müller-Stach, Xavier Roulleau, and Tomasz Szemberg

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Abstract

We study curves of negative self-intersection on algebraic surfaces. In contrast to what occurs in positive characteristics, it turns out that any smooth complex projective surface X with a surjective nonisomorphic endomorphism has bounded negativity (i.e., that C2 is bounded below for prime divisors C on X). We prove the same statement for Shimura curves on quaternionic Shimura surfaces of Hilbert modular type. As a byproduct, we obtain that there exist only finitely many smooth Shimura curves on such a surface. We also show that any set of curves of bounded genus on a smooth complex projective surface must have bounded negativity.

Article information

Source
Duke Math. J., Volume 162, Number 10 (2013), 1877-1894.

Dates
First available in Project Euclid: 11 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1373546606

Digital Object Identifier
doi:10.1215/00127094-2335368

Mathematical Reviews number (MathSciNet)
MR3079262

Zentralblatt MATH identifier
1272.14009

Subjects
Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
Secondary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]

Citation

Bauer, Thomas; Harbourne, Brian; Knutsen, Andreas Leopold; Küronya, Alex; Müller-Stach, Stefan; Roulleau, Xavier; Szemberg, Tomasz. Negative curves on algebraic surfaces. Duke Math. J. 162 (2013), no. 10, 1877--1894. doi:10.1215/00127094-2335368. https://projecteuclid.org/euclid.dmj/1373546606


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