A Computational View on the Non-degeneracy Invariant for Enriques Surfaces

IF 0.7 4区数学 Q2 MATHEMATICS

Experimental Mathematics Pub Date : 2022-02-03 DOI:10.1080/10586458.2022.2113576

Riccardo Moschetti, Franco Rota, L. Schaffler

引用次数: 3

Abstract

. For an Enriques surface S , the non-degeneracy invariant nd( S ) retains information on the elliptic fibrations of S and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on S together with a configuration of smooth rational curves, and gives a lower bound for nd( S ) . We provide a SageMath code that computes this combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying nd( S ) = 10 which are not general and with infinite automorphism group. We obtain lower bounds on nd( S ) for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes–Pardini. Finally, we recover Dolgachev and Kond¯o’s computation of the non-degeneracy invariant of the Enriques surfaces with finite automorphism group and provide additional information on the geometry of their elliptic fibrations.

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Enriques曲面的非退化不变量的计算观点

．对于Enriques曲面S，非简并不变量和(S)保留了关于S的椭圆纤摇及其极化的信息。本文引入了依赖于S的非退化不变量的一个组合形式和光滑有理曲线的一个构形，并给出了nd(S)的下界。我们提供了计算这个组合不变量的SageMath代码，并在几个示例中应用它。首先，我们确定了一类满足和(S) = 10的非一般且具有无限自同构群的结点Enriques曲面。我们得到了Mendes Lopes-Pardini研究的具有8条不相交光滑有理曲线的Enriques曲面的下界。最后，我们恢复了Dolgachev和Kond¯o对有限自同构群的Enriques曲面的非退化不变量的计算，并提供了关于其椭圆纤振几何的附加信息。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.