Samuel Grushevsky

Superstring Scattering Amplitudes in Higher Genus

In this paper we continue the program pioneered by D’Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ansätze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots.

The superstring cosmological constant and the Schottky form in genus 5

Combining certain identities for modular forms due to Igusa with Schottky-Jung relations, we study the cosmological constant for the recently proposed ansatz for the chiral superstring measure in genus 5. The vanishing of this cosmological constant turns out to be equivalent to the long-conjectured vanishing of a certain explicit modular form of genus 5 on the moduli of curves

Compactification of strata of Abelian differentials

{\cal M}_5

Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants

, and we disprove this conjecture, thus showing that the cosmological constant for the proposed ansatz does not vanish identically. We exhibit an easy modification of the genus 5 ansatz satisfying factorization constraints and yielding a vanishing cosmological constant. We also give an expression for the cosmological constant for the proposed ansatz that should hold for any genus if certain generalized Schottky-Jung identities hold.

The vanishing of two-point functions for three-loop superstring scattering amplitudes

We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.

The zero section of the universal semiabelian variety and the double ramification cycle

We prove that Prym varieties are characterized geometrically by the existence of a symmetric pair of quadrisecant planes of the associated Kummer variety. We also show that Prym varieties are characterized by certain (new) theta-functional equations. For this purpose we construct and study a difference-differential analog of the Novikov-Veselov hierarchy.

The class of the locus of intermediate Jacobians of cubic threefolds

In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen [2] vanishes. Our proof uses the reformulation of the ansatz given in [8], theta functions, and specifically the theory of the Γ00 linear system, introduced by van Geemen and van der Geer [6], on Jacobians.At the two-loop level, where the amplitudes were computed by D’Hoker and Phong [11–14, 17, 18], we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera [3, 8, 25].

Stable cohomology of the perfect cone toroidal compactification of g

We study the Chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties (ppav). We describe the subring generated by divisor classes, and compute the class of the partial compactification of the universal zero section, which turns out to lie in this subring. Our formula extends the results for the zero section of the universal uncompactified family. The partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of the moduli space of ppav. Our formula provides a first step in a program to understand the Chow groups of toroidal compactifications of the moduli of ppav, especially of the perfect cone compactification, by induction on genus. By restricting to the locus of Jacobians of curves, our results extend the results of Hain on the double ramification (two-branch-point) cycle.

Some intersection numbers of divisors on toroidal compactifications of _{ℊ}

We study the locus of intermediate Jacobians of cubic threefolds within the moduli space

Theta functions of arbitrary order and their derivatives

\mathcal{A}_{5}