Abelian surfaces of type (1,3) and quartic surfaces with 16 skew lines.

*(English)*Zbl 0809.14027The purpose of this article is to relate the geometry of certain moduli spaces of complex abelian surfaces with some quartic surfaces containing a certain configuration of lines. In particular, one of these moduli spaces is shown to be a Calabi-Yau threefold. Part of these results were also independently found by I. Naruki [Proc. Japan Acad., Ser. A 67, No. 7, 223-225 (1991; Zbl 0763.14019)]. The constructions go as follows. It was shown in Nieto’s thesis [see I. Nieto, “Invariante Quartiken unter der Heisenberg Gruppe \(T\)” (Erlangen 1989; Zbl 0668.14028)] that the Kummer surface of a general abelian surface with a type (2,6) polarization embeds (after blowing-up its 16 singular points) as a smooth quartic \(Q\) in \(\mathbb{P}^ 3\), invariant under the order-16 Heisenberg group \(H\). The 16 blown-up half-periods become 16 skew lines on \(Q\), which form an orbit under the action of \(H\).

There is a second \(H\)-orbit of 16 lines of \(Q\), corresponding to symmetric divisors \(\Theta\) such that \(2\Theta\) represents the polarization. In particular, this construction yields a birational correspondence between the moduli space of abelian surfaces with a symmetric line bundle of type (1,3) and a level-2 structure, and a certain subvariety \(M\) of the Grassmannian of lines in \(\mathbb{P}^ 3\) (those belonging to a 16-tuple as above). This map is \(H\)-equivariant and induces a birational correspondence between the moduli space of abelian surfaces with a polarization of type (1,3) and a level-2 structure, and the quotient \(M/H\). Here \(M/H\) parametrizes the sets of 16 lines as above. Since each quartic \(Q\) contains two such sets, \(M/H\) is an unbranched double cover of the variety \(N\) that parametrizes the quartics \(Q\) as above.

Explicit equations are given: in \(\mathbb{P}^ 5\), \(M\) is defined by \(\sum x_ i^ 2 = \sum 1/x_ i^ 2 = 0\) and \(N\) by \(\sum x_ i = \sum 1/x_ i = 0\). These allow the authors to prove that \(M\) is of general type, and that \(M/H\) is birational to a smooth Calabi-Yau threefold with Euler characteristic 80. Similarly, \(N\) is birational to a smooth Calabi- Yau threefold with Euler characteristic 100, whose Hodge numbers were calculated by Van Straten.

There is a second \(H\)-orbit of 16 lines of \(Q\), corresponding to symmetric divisors \(\Theta\) such that \(2\Theta\) represents the polarization. In particular, this construction yields a birational correspondence between the moduli space of abelian surfaces with a symmetric line bundle of type (1,3) and a level-2 structure, and a certain subvariety \(M\) of the Grassmannian of lines in \(\mathbb{P}^ 3\) (those belonging to a 16-tuple as above). This map is \(H\)-equivariant and induces a birational correspondence between the moduli space of abelian surfaces with a polarization of type (1,3) and a level-2 structure, and the quotient \(M/H\). Here \(M/H\) parametrizes the sets of 16 lines as above. Since each quartic \(Q\) contains two such sets, \(M/H\) is an unbranched double cover of the variety \(N\) that parametrizes the quartics \(Q\) as above.

Explicit equations are given: in \(\mathbb{P}^ 5\), \(M\) is defined by \(\sum x_ i^ 2 = \sum 1/x_ i^ 2 = 0\) and \(N\) by \(\sum x_ i = \sum 1/x_ i = 0\). These allow the authors to prove that \(M\) is of general type, and that \(M/H\) is birational to a smooth Calabi-Yau threefold with Euler characteristic 80. Similarly, \(N\) is birational to a smooth Calabi- Yau threefold with Euler characteristic 100, whose Hodge numbers were calculated by Van Straten.

Reviewer: O.V.Debarre (Strasbourg)

##### MSC:

14J10 | Families, moduli, classification: algebraic theory |

14K10 | Algebraic moduli of abelian varieties, classification |

14J30 | \(3\)-folds |