## 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

#### Abstract

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

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

Dates
First available in Project Euclid: 19 October 2016

https://projecteuclid.org/euclid.rmjm/1476884582

Digital Object Identifier
doi:10.1216/RMJ-2016-46-4-1263

Mathematical Reviews number (MathSciNet)
MR3563181

Zentralblatt MATH identifier
1365.14016

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1476884582

#### References

• V.V. Batyrev and O.N. Popov, The Cox ring of a del Pezzo surface, in Arithmetic of higher-dimensional algebraic varieties, B. Poonen and Y. Tschinkel, eds., Progr. Math. 226, Birkhäuser, Boston, 2004.
• M. Demazure, Surfaces de Del Pezzo I, II, III, IV, V, in Séminaire sur les singularités des surfaces, M. Demazure, H. Pinkham and B. Teissier, eds., Lect. Notes Math. 777, Springer-Verlag, New York, 1980.
• P. Du Val, On the directrices of a set of points in a plane, Proc. Lond. Math. Soc. 35 (1931), 23–74.
• A. Iqbal, A. Neitzke and C. Vafa, A mysterious duality, Adv. Theor. Math. Phys. 5 (2001), 769–807.
• J.H. Lee, Gosset polytopes in Picard groups of del Pezzo surfaces, Canad. J. Math. 64 (2012), 123–150.
• ––––, Configurations of lines in del Pezzo surfaces with Gosset polytopes, Trans. Amer. Math. Soc. 366 (2014), 4939–4967.
• N.C. Leung and J.J. Zhang, Moduli of bundles over rational surfaces and elliptic curves I: Simply laced cases, J. Lond. Math. Soc. 80 (2009), 750–770.
• Y. Manin, Cubic forms: Algebra, geometry, arithmetic, Nauka, Moscow, 1972; North-Holland, Amsterdam, 1974, second edition, 1986 (in English).
• L. Manivel, Configurations of lines and models of Lie algebras, J. Algebra 304 (2006), 457–486.