# Negative curves on algebraic surfaces

Th. Bauer, B. Harbourne, A. L. Knutsen, A. Küronya, S. Müller-Stach, T. Szemberg

Number | 2 |
---|---|

Author | Stefan Müller-Stach |

Year | 2012 |

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.