General birationality and hyperelliptic theta divisors

IF 1 3区 数学 Q1 MATHEMATICS
Fabrizio Catanese
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引用次数: 0

Abstract

We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of rational maps, via specializations. Among the applications is a new proof of the main result of Catanese and Cesarano (Electron Res Arch 29(6):4315–4325, 2021) that, for a general pair (AX) of an (ample) Hypersurface X in an Abelian Variety A, the canonical map \(\Phi _X\) of X is birational onto its image if the polarization given by X is not principal. The proof is also based on a careful study of the Theta divisors of the Jacobians of Hyperelliptic curves, and some related geometrical constructions. We investigate these here also in view of their beauty and of their independent interest, as they lead to a description of the rings of Hyperelliptic theta functions.

一般双性和超椭圆因子
我们首先陈述了一个条件,确保在图像上有一个双域映射是不可约正规非规则变种族的开放性质。然后给出一些标准,通过专门化来确保一组有理映射的一般双性性。其中一个应用是对Catanese和Cesarano(电子研究,29(6):4315-4325,2021)的主要结果的一个新的证明,即对于阿别变体a中的(样本)超曲面X的一般对(a, X),如果X给出的偏振不是主偏振,则X的正则映射\(\Phi _X\)在其像上是双象的。该证明还基于对超椭圆曲线雅可比矩阵的因子的仔细研究,以及一些相关的几何构造。我们在这里也考虑到它们的美丽和它们独立的兴趣,因为它们导致了对超椭圆函数环的描述。
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
CiteScore
2.10
自引率
10.00%
发文量
99
审稿时长
>12 weeks
期刊介绍: This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it). A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.
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