Degenerations of (1, 7)-polarized abelian surfaces 0 Introduction

The moduli space A(1, 7) of (1, 7)-polarized abelian surfaces with a level structure was shown by Manolache and Schreyer to be rational with compactification V (K 4 ) a Fano 3-fold V 22 More precisely, the (1, 7)-polarized abelian surface A with its level structure is embedded in P 6 = PV 0 , where V 0 is the Schrödinger representation of the Heisenberg group H 7 of level 7. The embedding is invariant under the action of G 7 , an extension of H 7 by an involution. The fixed points of this involution and its conjugates in G 7 form an H 7 -orbit of planes P + 2 and 3-spaces P − 3 . The 3-fold V (K 4 ) parameterizes, what we denote by generalized G 7 -embedded abelian surfaces (cf. 1.9). Every such surface intersects P + 2 in a finite subscheme of length six, which we may classify by its type, namely the length of its components. In this paper we prove the Theorem Let A be a generalized G 7 -embedded abelian surface of PV 0 , then according to the type of ζ A = A∩P + 2 the surface A is type of ζ A description (1, 1, 1, 1, 1, 1) smooth and abelian (2, 1, 1, 1, 1) translation scroll (E, ±σ) with 2 · σ = 0 (3, 1, 1, 1) tangent scroll (E, 0) (2, 2, 2) double translation scroll (E, ±σ) with 2 · σ = 0 and σ = 0 (2, 2, 1, 1) union of seven quadrics (4, 2) union of seven double projective planes (2, 2, 2) s union of fourteen projective planes For the translation and tangent scrolls, E is a septimic 1 , i.e. of degree seven, elliptic curve with an origin and the translation defined by the specified point σ. The two distinct (2, 2, 2) cases (abusively denoted by (2, 2, 2) and (2, 2, 2) s ) are described in In the first section we recall some basic facts on (1, 7)-polarized abelian surfaces with level structure, and construct a compactification of their moduli space. In fact, we consider a rational map κ : PV 0 / / P 6 defined by G 7 -invariant hypersurfaces of degree 7, which maps any general surface A ∈ A(1, 7) to a six-secant plane to a certain Veronese surface S ⊂ P 6 of degree 9. The surface S is the image by κ of P + 2 , so the six points * Partially supported by the European Commision through HPRN-CT-2000-00099