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Even sets of (\(-4\))-curves on rational surface. (English) Zbl 1229.14030

An even set of \((-r)\)-curves on a smooth complex projective algebraic surface is a set of smooth irreducible disjoint rational curves \(C_1, \dots, C_n\) with self-intersection \(-r\) such that the linear equivalence class of the divisor \(C:=C_1+ \dots +C_n\) is divisible by \(2\) in the Picard group. The author studies rational surfaces \(X\) containing an even set of \((-4)\)-curves through the double cover \(\pi:S \to X\) branched along \(C\). Up to contracting \((-1)\)-curves on \(X\) disjoint from \(C\), she proves that \(S\) is minimal and \(S\) is either a \(K3\) surface, or a surface of Kodaira dimension \(\kappa(S)=1\) or \(2\). In every case it turns out that \(n\) is bounded, unlike what happens for even sets of \((-2)\)-curves. She proves that \(\kappa(S) \leq 1\) if and only if \(X\) is endowed with a not relatively minimal elliptic fibration, containing \(C_1, \dots , C_n\) in its singular fibers. Moreover, examples are produced for all possible values of \(n\) when \(X\) is a \(K3\) surface (\(1 \leq n \leq 10\) in this case).

MSC:

14J26 Rational and ruled surfaces
14J17 Singularities of surfaces or higher-dimensional varieties
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14C20 Divisors, linear systems, invertible sheaves
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