Negative curves on algebraic surfaces

By Thomas Bauer, Brian Harbourne, A. Knutsen, Alex Kuronya, S. Müller–Stach, X. Roulleau, T. Szemberg. 2011

82 citations recorded by the source corpus.

We study curves of negative self-intersection on algebraic surfaces. Our main result shows there exist smooth complex projective surfaces X, related to Hilbert modular surfaces, such that X contains reduced, irreducible curves C of arbitrarily negative self-intersection C 2 . Previously the only known examples of surfaces for which C 2 was not bounded below were in positive characteristic, and the general expectation was that no examples could arise over the complex numbers. Indeed, we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field, and thus our complex examples require a different approach.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d76cebd612f6a429ec878b93cc7f9c98bf2d4122

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Tomasz Szemberg
2. Brian Harbourne
3. Thomas Bauer
4. Jack Huizenga
5. Sandra Di Rocco
6. Cristiano Bocci

In Mathematical Physics

1. Brian Harbourne
2. Jack Huizenga
3. Tomasz Szemberg

Citations in the corpus

• Bounded Negativity and Arrangements of Lines (2014, background): “In those cases where the mi are all equal, we of course have mi = m = m, but (over the complex numbers) we know of only one such nontrivial configuration with m > 2, and hence m = 3 by the Hirzebruch inequality (3).”