Rocky Mountain Journal of Mathematics

Contractions of del Pezzo surfaces to $\mathbb P^2$ or $\mathbb P^1\times \mathbb P^1$

Jae-Hyouk Lee

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In this article, we consider $r-1$ disjoint lines given in a del~Pezzo surface $S_{r}$ and study how to determine if a contraction given by the lines produces a map to $S_{1}$ (one point blow up of $\mathbb {P}^{2}$) or $\mathbb {P}^{1}\times \mathbb {P}^{1}$ by checking only the configuration of lines. Here, we show that we can determine if the disjoint lines produce a contraction to $\mathbb {P}^{1}\times \mathbb {P}^{1}$ by combining a quartic rational divisor class to them. We also study the quartic rational divisor classes along the configuration of lines in del Pezzo surfaces.

Article information

Rocky Mountain J. Math., Volume 46, Number 4 (2016), 1263-1273.

First available in Project Euclid: 19 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E05: Rational and birational maps 14J26: Rational and ruled surfaces

del Pezzo surface Gosset polytopes Weyl action


Lee, Jae-Hyouk. Contractions of del Pezzo surfaces to $\mathbb P^2$ or $\mathbb P^1\times \mathbb P^1$. Rocky Mountain J. Math. 46 (2016), no. 4, 1263--1273. doi:10.1216/RMJ-2016-46-4-1263.

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