论文标题
theta表面
Theta surfaces
论文作者
论文摘要
仿射3空间中的theta表面是3属中的riemann theta函数的零集。这包括由特殊平面四分之一引起的表面，这些表面是单数或还原的特殊平面。 Lie和Poincaré表明，Theta表面恰恰是双翻译的表面，即以两种不同的方式作为两个空间曲线的Minkowski总和获得。这些曲线是通过Abelian积分参数化的，因此它们通常不是代数。本文通过计算视角提供了有关此古典主题的新观点。我们提出了在四分之一曲线及其theta表面之间传递的实用工具，并开发了theta函数退化的数值代数几何。
A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that theta surfaces are precisely the surfaces of double translation, i.e. obtained as the Minkowski sum of two space curves in two different ways. These curves are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.
