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Negative curves on algebraic surfaces. (English) Zbl 1272.14009
Let \(X\) be a smooth complex projective surface. A classical but nevertheless completely open conjecture claims that there exists a positive integer \(b(X)\) such that for every integral curve \(C \subset X\), the self-intersection number \(C^2\) is bounded below by \(-b(X)\). The goal of this paper is to give some evidence in favour of this conjecture: a natural strategy for a counterexample is to consider a surface \(X\) admitting an endomorphism, i.e., a surjective map \(X \rightarrow X\) of degree at least two, and study the pull-back of a curve with negative self-intersection. However the authors prove that the bounded negativity conjecture holds for every surface admitting an endomorphism. A more general strategy is to consider a surface with an interesting correspondence, for example a Hecke correspondence on a quaternionic Shimura surface of general type (we refer to the paper for precise definitions). Hecke correspondence are known to preserve the set of Shimura curves on the surface, yet again the authors prove that Shimura curves verify the bounded negativity conjecture. Finally the authors prove a weak form of the conjecture if \(X\) is a non-uniruled surface: let \(X\) be surface of non-negative Kodaira dimension. Then there exists a positive integer \(b(X,g)\) such that \(C^2 \geq - b(X,g)\) for every integral curve \(C \subset X\) of geometric genus at most \(g\).

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
14G35 Modular and Shimura varieties
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