We prove that a 1-dimnl family of abelian varieties with an ample sheaf
defining principal polarization can be canonically compactified (after a finite
base change) to a projective family with an ample sheaf. We show that the
central fiber (P,L), which we call an SQAV, has a canonical Cartier theta
divisor. We give a combinatorial definition for SQAVs and describe their
geometrical properties, in particular compute cohomologies of L^n, n\ge0.Comment: Final version, to appear in Tohoku Math.
