Abelian surfaces of type \((1,4)\)
Let \(A\) be an abelian surface over \({\mathbb{C}}\) and \(L\) a (symmetric) polarization on \(A\) of type \((1,4)\). Let \(K\) be the kernel of the natural map of \(A\) to the dual of \(A\) and \(K\) decomposes into a direct sum of two isotropic cyclic subgroups of order \(4\cdot K=K_ 1\oplus K_ 2\). The authors study the rational map (regular in fact) determined by the linear system of the polarization. This was the only remaining case to be studied for polarized abelian surfaces. - The main results are the following: (i) (Theorem I (i)) The map \(\phi_ L: A\to \bar A\subset {\mathbb{P}}_ 3\) is birational onto a singular octic \(\bar A\) in \({\mathbb{P}}_ 3\) iff the polarization divisor admits an involution which commutes with the translation with \(K_ 2\), and whose quotient is an elliptic curve. (ii) (Theorem I (ii)) In the exceptional case \(\phi_ L\) is a double covering of a singular quartic \(\bar A\) which is birational to an elliptic scroll. (iii) (Theorem II) One can write down the octic equation of \(\bar A\) explicitly with 4 homogeneous parameters and this determines a finite rational map of \({\mathbb{P}}_ 3\) to the moduli space of polarized abelian surface of type \((1,4)\) of degree 48.
moduli space of polarized abelian surface
scientific article; zbMATH DE number 4175145