Theta functions of totally degenerate curves
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.
scientific article; zbMATH DE number 4216215