A generalization of the Chowla-Selberg formula.
A generating function for fatgraphs
A geometric approach to the Jacobian Conjecture in ℂ²
We consider polynomial mappings (f,g) of ℂ² with constant nontrivial jacobian. Using the Riemann-Hurwitz relation we prove among other things the following: If g - c (resp. f - c) has at most two branches at infinity for infinitely many numbers c or if f (resp. g) is proper on the level set (resp. ), then (f,g) is bijective.
A geometric description of the sum of linear series of order two on a curve of genus 2.
A Geometrie Approach to Keller's Jacobian Conjecture.
A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence
We show how the Landau–Ginzburg/Calabi–Yau correspondence for the quintic three-fold can be cast into a global mirror symmetry framework. Then we draw inspiration from Berglund–Hübsch mirror duality construction to provide an analogue conjectural picture featuring all Calabi–Yau hypersurfaces within weighted projective spaces and certain quotients by finite abelian group actions.
A group action on Losev-Manin cohomological field theories
We discuss an analog of the Givental group action for the space of solutions of the commutativity equation. There are equivalent formulations in terms of cohomology classes on the Losev-Manin compactifications of genus moduli spaces; in terms of linear algebra in the space of Laurent series; in terms of differential operators acting on Gromov-Witten potentials; and in terms of multi-component KP tau-functions. The last approach is equivalent to the Losev-Polyubin classification that was obtained...
A group law on smooth real quartics having at least real branches
Let be a smooth real quartic curve in . Suppose that has at least real branches . Let and let . Let be the map from into the neutral component Jac of the set of real points of the jacobian of , defined by letting be the divisor class of the divisor . Then, is a bijection. We show that this allows an explicit geometric description of the group law on Jac. It generalizes the classical geometric description of the group law on the neutral component of the set of real points of...
A Higher Dimensional Analogue of Mordell's Conjecture Over Function Fields.
A Lax formalism for the elliptic difference Painlevé equation.
A limit linear series moduli scheme
We develop a new, more functorial construction for the basic theory of limit linear series, which provides a compactification of the Eisenbud-Harris theory. In an appendix, in order to obtain the necessary dimensional lower bounds on our limit linear series scheme we develop a theory of “linked Grassmannians”; these are schemes parametrizing sub-bundles of a sequence of vector bundles, which map into one another under fixed maps of the ambient bundles.
A look into the Severi varieties of curves in higher codimension.
A metric graph satisfying [...] w 4 1 = 1 that cannot be lifted to a curve satisfying [...] dim ( W 4 1 ) = 1
For all integers g ≥ 6 we prove the existence of a metric graph G with [...] w41=1 such that G has Clifford index 2 and there is no tropical modification G′ of G such that there exists a finite harmonic morphism of degree 2 from G′ to a metric graph of genus 1. Those examples show that not all dimension theorems on the space classifying special linear systems for curves have immediate translation to the theory of divisors on metric graphs.
A modular compactification of the general linear group.
A Narasimhan-Seshardri-Donaldson Correspondance over Non-orientable Surfaces.
A New Characterization of the Integer 5906.
A new compactification of the moduli space of curves
A new construction of -adic -functions attached to certain elliptic curves with complex multiplication
In this paper we apply the results of our previous article on the -adic interpolation of logarithmic derivatives of formal groups to the construction of -adic -functions attached to certain elliptic curves with complex multiplication. Our results are primarily concerned with curves with supersingular reduction.
A new cubic character sum
A normal form for curves in Grassmannians.