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Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space A_6 of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general ppav of dimension 6 is a Prym-Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_6 lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_6-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge-Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym-Tyurin map from the Hurwitz space to A_6 in the terms of syzygies of the Abel-Prym-Tyurin curve.
We give a characterizaton of smooth ample Hypersurfaces in Abelian Varieties and also describe an irreducible connected component of their moduli space: it consists of the Hypersurfaces of a given polarization type, plus the iterated univariate cover
We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${mathcal A}_g$, satisfying certain additivity and finiteness properties, in term
Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron (who treated the case $e=14$), we prove that the moduli space of polarized K3 surfaces of degree $2e$ contains complete curves for all $egeq 62$ and for some sporadic lower valu
Let $mathcal{M}(n,m;F bp^n)$ be the configuration space of $m$-tuples of pairwise distinct points in $F bp^n$, that is, the quotient of the set of $m$-tuples of pairwise distinct points in $F bp^n$ with respect to the diagonal action of ${rm PU}(1,n;
We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with normal crossin