Projective normality of abelian surfaces given by primitive line bundles
Let \(A\) be an abelian variety with \(\dim A=g\) and let \(L\) be an ample line bundle of type \(\delta=(\delta_{1},\dots , \delta_{g})\) on \(A\). Then the embedding of \(A\) given by the morphism defined by \(L^{n}\) is projectively normal if \(n\geq 3\) or \(n=2\) and no point of \(K(L^{2})\) is a base point for \(L\), where \(K(L^{2})=\{a\in A \mid L\cong t_{a}^{*}L\}\) and \(t_{a}\) is the translation map by \(a\) on \(A\). In this paper, the author studies the case of an abelian surface when \(L\) is primitive, that is, \(L\) is of the type \((1, \delta)\). If \(\delta\leq 4\), then \(L\) is not very ample. If \(\delta=5\) or \(6\), then the embedding defined by the linear system \(|L|\) is not projectively normal. So the author considers the case where \(\delta\geq 7\) and obtains the following theorem: Let \(L\) be an ample line bundle of type \((1, \delta)\) on an abelian surface \(A\). Suppose the Néron-Severi group of \(A\) is isomorphic to \(\mathbb{Z}\) and \(\delta\geq 7\). Then the image of \(A\), under the morphism defined by the linear system \(|L|\), is projectively normal. This case is the generic situation of \textit{G. P. Pirola} [Math. Ann. 282, No. 3, 361-368 (1988; Zbl 0625.14024)]. This result can be proved by an analysis of the representations of the theta group \(\mathcal{G}(L)\) on \(\text{Sym}^{2}H^{0}(L)\) as well as on \(H^{0}(L^{2})\).
scientific article; zbMATH DE number 1275769
scientific article