Theta functions of totally degenerate curves
type: Item
instance of: scholarly article
title: Theta functions of totally degenerate curves
zbMATH Open document ID: 0739.14017
DOI: 10.1007/BF01197329
published in: Archiv der Mathematik
publication date: 1992-01-01T00:00:00Z
review text: If \(C\) is an irreducible, stable, totally degenerate curve over a field \(k\) there exist \(g\)-tupels \(a=(a_ 1,\ldots,a_ g)\), \(b=(b_ 1,\ldots,b_ g)\) of points on the projective line such that \(C\) is obtained from \(\mathbb{P}_ 1\times k\) by identifying the points \(a_ i\), \(b_ i\) for all \(i\). The pair \((a,b)\) determines a canonical base \(\omega_ 1,\ldots,\omega_ g\) of differentials of the first kind on \(C\) and each \(\omega_ i\) is of the form \(du_ i/u_ i\) where \(u_ i=u_ i(z)\) is a rational function on \(\mathbb{P}_ 1\times k\) whose divisor is \((a_ i)-(b_ i)\). The period matrix \(\hbox{per}(u):=u_ i(a_ j)/u_ i(b_ j), 1\leq i,j\leq g\) is symmetric and its diagonal is zero. The totally degenerate theta function \(\theta(x_ 1,\ldots,x_ g)\) relative to \(\hbox{per}(u)\) is a polynomial in \(g\) variables \(x_ 1,\ldots,x_ g\). Thus \(\theta(u_ 1(z),\ldots,u_ g(z))\) is a rational function on \(C\). Its divisor is called a theta divisor \(D_ u\) on \(C\), if the support of \(D_ u\) does not contain any of the points \(a_ i\), \(b_ i\). It is a divisor of degree \(g\). One gets the analogue of a fundamental theorem of Riemann: there is a constant \(\rho(u)\), \(u:=(u_ 1,\ldots,u_ g)\) such that \(\varphi^ g_ u(D_{cu}) = {\rho(u)\over c}\), if \(D_{cu}\) is a divisor on \(C\), where \(\varphi^ g_ u\) is the Abel product map from the set of divisors of degree \(g\) onto the algebraic torus \(\mathbb{G}^ g_ m\times k\). The Riemann constant \(\rho(u)\) is determined as a function of the moduli parameters \(a_ i\), \(b_ i\). As a corollary one gets the Riemann vanishing theorem.
Mathematics Subject Classification ID: 14H42
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 32G20
zbMATH DE Number: 4216215
zbMATH Keywords: totally degenerate theta function
zbMATH Keywords: theta divisor
zbMATH Keywords: Riemann vanishing theorem
zbMATH Keywords: period matrix
author: Lothar Gerritzen
MaRDI profile type: Publication
cites work: On the algebra of totally degenerate theta functions
cites work: Tata lectures on theta. II: Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura
full work available at URL: https://doi.org/10.1007/bf01197329
OpenAlex ID: W2021804768
description: scientific article; zbMATH DE number 4216215