In this article, we consider r - 1 disjoint lines given in a del Pezzo surface Sr and study how to determine if a contraction given by the lines produces a map to S1 (one point blow up of ℙ2) or ℙ1 x ℙ1 by checking only the configuration of lines. Here, we show that we can determine if the disjoint lines produce a contraction to ℙ1 x ℙ1 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.
Bibliographical noteFunding Information:
This study was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) and funded by the Ministry of Education, Science and Technology (grant No. NRF-2013R1A1A2012783).
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- Del Pezzo surface
- Gosset polytopes
- Weyl action