Heisenberg-invariant Kummer Surfaces Ii

is the closure of the locus parametrizing H22-invariant quartics with 16 skew lines. The smooth surfaces of this type are parametrized by a non-empty open set N s of N . These surfaces are Kummer surfaces associated to abelian surfaces with a (1,3)–polarization. The action of the Heisenberg group on the Kummer surface corresponds to a level-2 structure on the abelian surface. If A is an abelian surface and L is a symmetric line bundle representing this polarization then L⊗2 is the unique totally symmetric line bundle representing the (2,6)–polarization. Under the involution ι : x 7→ −x the space H0(L⊗2) decomposes into eigenspaces H0(L⊗2)+ and H0(L⊗2)− of dimension 8 and 4 respectively. The linear system given by H0(L⊗2)− defines a rational map A − − → P3 which induces a map from the smooth Kummer surface K̃m(A) to P3. In fact for general A this defines an embedding of K̃m(A) into P3. With respect to a suitable basis of the linear system which defines the map to P3 the image surface is H22-invariant. Let A1,3(2) be the moduli space of (1, 3)-polarized abelian surfaces with a level-2 structure. Then the map which associates to an abelian surface its Kummer surface defines a 2 : 1 map A1,3(2) → N (see also part I [HNS] of this paper). The complement of N s in N consists of two sets of 15 planes called the Vand S-planes respectively. In part I we showed that the Kummer surfaces which are parametrized by the V-planes are exactly the Kummer surfaces of bielliptic abelian surfaces. In this part we consider the quartic surfaces which are parametrized by the 15 S-planes and we will show that they correspond to Kummer surfaces of degenerate abelian surfaces.