Algebraic Geometry Seminar: Vivek Shende

A divisor on a curve is called "special'' if its linear equivalence class is larger than expected.  On a hyperelliptic curve, all such come from pullbacks of points from the line.  But one can ask subtler questions.  Fix a degree zero divisor Z; consider the space parameterizing divisors D where D and D+Z are both special.  In other words, we wish to study the intersection of the theta divisor with a translate; the main goal is to understand its singularities and its cohomology.

The real motivation comes from number theory.  Consider, in products of the moduli space of elliptic curves, points whose coordinates all correspond to curves with complex multiplication.  The Andre-Oort conjecture controls the Zariski closure of sequences of such points (and in this case is a theorem of Pila) and a rather stronger equidistribution statement was conjectured by Zhang.  The locus introduced above arises naturally in the consideration of a function field analogue of this conjecture.  This talk presents joint work with Jacob Tsimerman.