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arXiv 2003.017752020-03math.AG

用 Macaulay2 计算 A1-Euler 数

Computing A1-Euler numbers with Macaulay2

Sabrina Pauli

备注：: Submitted to Research in the Mathematical Sciences: Special Issue 'S.I. : Arithmetic Topology - PIMS 2019'
发表信息：: Res. Math. Sci. 10, No. 3, Paper No. 26, 14 p. (2023)
DOI：: 10.1007/s40687-023-00392-0

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AI总结：

本文利用 Macaulay2 在 GW(k) 中进行富集计数，重新证明三次曲面直线数为 3+12h，并计算多种几何计数及 A1-Milnor 数。

摘要：

我们使用 Macaulay2 对 GW(k) 中的若干富集计数进行计算。首先，我们在 GW(Fp) 中对素数 p 使用 Macaulay2 在 Fp 上计算一般三次曲面上的直线数，并在 GW(Q) 中对有理数 Q 进行计算。这为三次曲面上的直线数在 GW(k) 中等于 3+12h 这一事实提供了一个新证明，其中 h 表示双曲形式。然后，我们计算 P3 中与 4 条一般直线相交的直线数、二次曲面上与一条一般直线相交的直线数，以及 d 次曲面束中奇异元素的计数。最后，我们提供计算 EKL-形式的代码，并计算若干 A1-Milnor 数。

英文摘要：

We use Macaulay2 for several enriched counts in GW(k). First, we compute the count of lines on a general cubic surface using Macaulay2 over Fp in GW(Fp) for p a prime number and over the rational numbers Q in GW(Q). This gives a new proof for the fact that the count of lines on a cubic surface is 3+12h in GW(k) where h denotes the hyperbolic form. Then, we compute the count of lines in P3 meeting 4 general lines, the count of lines on a quadratic surface meeting one general line and the count of singular elements in a pencil of degree d-surfaces. Finally, we provide code to compute the EKL-form and compute several A1-Milnor numbers.

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