## 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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### References:

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