Asymptotic period relations for Jacobian elliptic surfaces

We find an asymptotic description of the period locus of simply connected Jacobian elliptic surfaces and of the period locus of hyperelliptic curves. The two descriptions are essentially the same, and are given by the alkanes of organic chemistry.Comment: The use of the Minimal Model Program has been clarified, as has the Torelli theorem for special elliptic surface

Siegel modular forms and the gonality of curves

There is no stable Siegel modular form that vanishes on the trigonal locus in every moduli space of curves.Comment: This version takes into account Yamada's correction of Fay's formula for the period matrix of a certain degenerating family of curve

"Del Pezzo surfaces as Springer fibres for exceptional groups"

We show that simultaneous log resolutions of simply elliptic singularities can be constructed inside suitable stacks of principal bundles over elliptic curves. In particular, we give a direct geometrical construction of del Pezzo surfaces from the corresponding exceptional simple algebraic groups.Comment: This is a re-written version of "From exceptional groups to del Pezzo surfaces and simultaneous log resolutions via principal bundles over elliptic curves". It contains 2 figures. This version corrects one of the figure

Moduli and periods of simply connected Enriques surfaces

We describe a period map for those simply connected Enriques surfaces in characteristic 2 whose canonical double cover is K3. The moduli stack for these surfaces has a Deligne-Mumford quotient that is an open substack of a $\mathbb P^1$-bundle over the period space. We also give some general results relating local and global moduli for algebraic varieties and describe the difference in their dimensions in terms of the failure of the automorphism group scheme to be reduced