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Quotients of del Pezzo Surfaces of Degree 2
Moscow Mathematical Journal. 2018. Vol. 18. No. 3. P. 557-597.
Let 𝕜 be any field of characteristic zero, X be a del Pezzo surface of degree 2 and G be a group acting on X. In this paper we study 𝕜-rationality questions for the quotient surface X/G. If there are no smooth 𝕜-points on X/G then X/G is obviously non-𝕜-rational. Assume that the set of smooth 𝕜-points on the quotient is not empty. We find a list of groups such that the quotient surface can be non-𝕜-rational. For these groups we construct examples of both 𝕜-rational and non-𝕜-rational quotients of both 𝕜-rational and non-𝕜-rational del Pezzo surfaces of degree 2 such that the G-invariant Picard number of X is 1. For all other groups we show that the quotient X/G is always 𝕜-rational.
Keywords: del Pezzo surface
Publication based on the results of:
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Trepalin A., Loughran D., / Cornell University. Series arXiv "math". 2019.
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Added: December 2, 2018
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Added: December 3, 2013
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Added: October 21, 2018
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Serge Lvovski, / Cornell University. Series arXiv "math". 2017.
We show that the monodromy group acting on $H^1(\cdot,\mathbb Z)$ of a smooth
hyperplane section of a del Pezzo surface over $\mathbb C$ is the entire
group $\mathrm{SL}_2(\mathbb Z)$. For smooth surfaces with $b_1=0$ and hyperplane section
of genus $g>2$, there exist examples in which a similar assertion is
false. Actually, if hyperplane sections of ...
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Loginov K., Moscow Mathematical Journal 2018 Vol. 18 No. 4 P. 721-737
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Yuri Prokhorov, / Cornell University. Series math "arxiv.org". 2011.
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Trepalin A., Central European Journal of Mathematics 2014
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