Abstract
Starting from a beautiful idea of Kanev, we construct a uniformization
of the moduli space
đť’ś
6
\mathcal{A}_{6}
of principally polarized abelian 6-folds in
terms of curves and monodromy data. We show that the general principally polarized abelian variety 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
E_{6}
lattice. Along the way, we establish numerous facts concerning the
geometry of the Hurwitz space of such
E
6
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
đť’ś
6
\mathcal{A}_{6}
in the
terms of syzygies of the Abel–Prym–Tyurin curve.
