摘要翻译:
属3的具有二级结构的非奇异完备超椭圆曲线的模簇是一个允许若干标准紧致的5维拟射影簇。第一个是X,它是把这个变体作为第2层和第3属的Siegel模变体的一个子变体来实现的。我们将用一个合适的射影嵌入描述X的方程组及其Hilbert函数。结果是X是正常的。另一个模型来自于几何不变理论,使用了(p^1)^8中所谓的半稳定退化点构型。我们用Y来表示这种git-紧性。在一个合适的射影嵌入中,这种变化的方程是已知的。这一变种也可以用Baily-Borel紧实球商来确定。我们将详细地描述这些结果,并得到新的证明,包括一些更精细的结果。我们有一个Y和X之间的出生图。本文利用分次代数(与模形式的代数密切相关)A,B这样x=proj(A)和y=proj(B)的事实。这种同态建立在Thomae(19世纪)的理论基础上,在Thomae的理论中,超椭圆曲线的thetanullwerte已经被计算出来。利用a,B$的显式方程,我们可以计算映射从Y到X的基轨迹,将基轨迹和Y的奇异性炸掉,我们得到一个占优的光滑模型{\tilde Y}。我们将看到{\tilde Y}与标记射影线族(p^1,x_1,...,x_8)的紧致同构,通常用{\bar M_{0,8}}表示。模型X和模型Y之间有几个组合相似性,如果用球模型来描述Y,这些相似性可以得到最好的描述。
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英文标题:
《The modular variety of hyperelliptic curves of genus three》
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作者:
E. Freitag, R.Salvati Manni
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最新提交年份:
2007
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分类信息:
一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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一级分类:Mathematics 数学
二级分类:Commutative Algebra 交换代数
分类描述:Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环,模,理想,同调代数,计算方面,不变理论,与代数几何和组合学的联系
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英文摘要:
The modular variety of non singular and complete hyperelliptic curves with level-two structure of genus 3 is a 5-dimensional quasi projective variety which admits several standard compactifications. The first one, X, comes from the realization of this variety as a sub-variety of the Siegel modular variety of level two and genus three .We will be to describe the equations of X in a suitable projective embedding and its Hilbert function. It will turn out that X is normal. A further model comes from geometric invariant theory using so-called semistable degenerated point configurations in (P^1)^8 . We denote this GIT-compactification by Y. The equations of this variety in a suitable projective embedding are known. This variety also can by identified with a Baily-Borel compactified ball-quotient. We will describe these results in some detail and obtain new proofs including some finer results for them. We have a birational map between Y and X . In this paper we use the fact that there are graded algebras (closely related to algebras of modular forms) A,B such that X=proj(A) and Y=proj(B). This homomorphism rests on the theory of Thomae (19th century), in which the thetanullwerte of hyperelliptic curves have been computed. Using the explicit equations for $A,B$ we can compute the base locus of the map from Y to X. Blowing up the base locus and the singularity of Y, we get a dominant, smooth model {\tilde Y}. We will see that {\tilde Y} is isomorphic to the compactification of families of marked projective lines (P^1,x_1,...,x_8), usually denoted by {\bar M_{0,8}}. There are several combinatorial similarities between the models X and Y. These similarities can be described best, if one uses the ball-model to describe Y.
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PDF链接:
https://arxiv.org/pdf/0710.5920