Contractions of del Pezzo surfaces to \(\mathbb P^2\) or \(\mathbb P^1\times \mathbb P^1\). (English) Zbl 1365.14016

A del Pezzo surface is a smooth projective surface \(S\) such that its anticanonical divisor \(-K_S\) is ample. The del Pezzo surfaces are classified and they are either the blow up \(S_r\) of \(\mathbb{P}^2\) in \(0\leq r\leq 8\) points or \(\mathbb{P}^1\times \mathbb{P}^1\). Therefore, if the Picard number of a del Pezzo surface \(S\) is 2, either \(S=\mathbb{P}^1\times \mathbb{P}^1\) or \(S=Bl_p \mathbb{P}^2\).
Let \(\varepsilon: S_r\rightarrow S\) be a birational morphism given by the contraction of \(r-1\) distinct \((-1)\)-curves. The paper under review gives a criterion to determine wether \(S=\mathbb{P}^1\times \mathbb{P}^1\) or \(S=Bl_p \mathbb{P}^2\). The main theorem affirms that, if there is a rational quartic \(q\) in \(S_r\) such that \[ 2q+K_{S_r}\equiv l_1+\ldots+ l_{r-2} \] where \(l_i\) are distinct \((-1)\)-curves, then the contraction of the \(l_i\) is \(\mathbb{P}^1\times \mathbb{P}^1\).
A more precise description of the quartics that appear in this way is given using the Gosset polytope of \(S_r\) studied by the author in [Trans. Am. Math. Soc. 366, No. 9, 4939–4967 (2014; Zbl 1346.14098); Can. J. Math. 64, No. 1, 123–150 (2012; Zbl 1268.14038)]. The vertices of the Gosset polytope of \(S_r\) correspond bijectively to the \((-1)\)-curves in \(S_r\). The birational morphisms given by the contraction of \(d\) disjoint \((-1)\)-curves correspond to \((d-1)\)-simplexes. Thus, if \(\varepsilon: S_r\rightarrow S\) is a birational morphism such that the Picard number of \(S\) is 2, then \(S=Bl_p \mathbb{P}^2\) if and only if the \((r-2)\)-simplex of the contraction is contained in an \((r-1)\)-simplex, that is, if there exist another \((-1)\)-curve disjoint from the first \(r-1\).


14E05 Rational and birational maps
14J26 Rational and ruled surfaces
14J45 Fano varieties
Full Text: DOI Euclid


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