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Computations and Moduli Spaces for Non-archimedean Varieties.

机译：非档案类品种的计算和模空间。

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摘要

Tropical geometry and non-archimedean analytic geometry study algebraic varieties over a field K with a non-archimedean valuation. One of the major goals is to classify varietiess over K by intrinsic tropical properties. This thesis will contain my work at UC Berkeley and my joint work with others towards the goal.;Chapter 2 discusses some moduli spaces and their tropicalizations. The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.;Chapter 3 develops numerical algorithms for Mumford curves over the field of p-adic numbers. Mumford curves are foundational to subjects dealing with non-archimedean varieties, and it has various applications in number theory. We implement algorithms for tasks such as: approximating the period matrices of the Jacobians of Mumford curves; computing the Berkovich skeleta of their analytifications; and approximating points in canonical embeddings.;Chapter 4 discusses how to tropicalize del Pezzo surfaces of degree 5, 4 and 3. A generic cubic surface P3 is a Del Pezzo surface of degree 3, which is obtained by blowing up the plane at 6 points. We study its tropicalization by taking the intrinsic embedding of the surface surface minus its 27 lines. Our techniques range from controlled modifications to running gfan on the universal Cox ideal over the relevant moduli space. We classify cubic surfaces by combinatorial properties of the arrangement of 27 trees obtained from the image of the 27 lines under this tropicalization.;Chapter 5 discusses the classical Cayley-Bacharach theorem, which states that if two cubic curves on the plane intersect at 9 points, then the 9th point is uniquely determined if 8 of the points are given. The chapter derives a formula for the coordinates of the 9th point in terms of the coordinates of the 8 given points. Furthermore, I will discuss the geometric meaning of the formula, and how it is related to del Pezzo surfaces of degree 3.
机译：热带几何和非阿基米德解析几何研究具有非阿基米德估值的K域上的代数变体。主要目标之一是通过热带固有属性对K上的品种进行分类。本文将包含我在加州大学伯克利分校的工作以及我与其他人为实现该目标而进行的共同工作。第二章讨论了模量空间及其热带化。可以使用拟阵理论对组合在单项式图下的超平面排列的补体图像进行热带化处理。我们将其应用于与复杂反射布置相关的经典模空间。从模块化曲线开始，我们访问Segre三次方，Igusa四次方和标记为2和3度的del del Pezzo曲面的模。我们的主要示例是Burkhardt四次方，其热带化是在39维空间中的3维扇形。这实现了对属2曲线的热带模量的具体和抽象方法的综合。;第三章为p-adic数域上的Mumford曲线开发了数值算法。 Mumford曲线是处理非Archededean变体的主题的基础，并且在数论中具有多种应用。我们为以下任务实现算法：近似Mumford曲线的Jacobian曲线的周期矩阵；计算Berkovich skeleta的分析结果；第四章讨论了如何对度为5、4和3的del Pezzo曲面进行热带化。通用三次曲面P3为度为3的Del Pezzo曲面，是通过将平面上的6个点炸开而获得的。我们通过减去27条线的表面固有嵌入来研究其热带化。我们的技术范围从受控修改到在相关模空间上在通用Cox理想系统上运行gfan。我们通过在这种热带化条件下从27条线的图像中获得的27棵树的排列的组合性质对立方表面进行分类。;第5章讨论了经典的Cayley-Bacharach定理，该定理指出，如果平面上的两条立方曲线在9个点处相交，则如果给出8个点，则唯一确定第9个点。本章根据8个给定点的坐标得出第9点坐标的公式。此外，我将讨论公式的几何含义，以及该公式与3级del Pezzo曲面的关系。

著录项

  • 作者

    Ren, Qingchun.;

  • 作者单位

    University of California, Berkeley.;

  • 授予单位 University of California, Berkeley.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2014
  • 页码 147 p.
  • 总页数 147
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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