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We show that the counting of rational curves on a complete toric variety which are in general position relative to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory
We describe a systematic way of constructing effective divisors on the moduli space of stable curves having exceptionally small slope. We show that every codimension 1 locus in consisting of curves failing to satisfy a Green-Lazarsfeld syzygy-type condition provides a counterexample to the Harris-Morrison slope conjecture. We also introduce a new geometric stratification of the moduli space of curves somewhat similar to the classical stratification given by gonality but where the analogues of hyperelliptic curves are the sections of surfaces
We construct a family of -symmetric biorthogonal abelian functions generalizing Koornwinder's orthogonal polynomials (see ) and prove a number of their properties, most notably analogues of Macdonald's conjectures. The construction is based on a direct construction for a special case generalizing Okounkov's interpolation polynomials (see ). We show that these interpolation functions satisfy a collection of generalized hypergeometric identities, including new multivariate elliptic analogues of Jackson's summation and Bailey's transformation
We prove the following conjecture recently formulated by Jakobson, Nadirashvili, and Polterovich (see [15, Conjecture 1.5, page 383]). On the Klein bottle , the metric of revolution , , is the unique extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof leads us to study a Hamiltonian dynamical system that turns out to be completely integrable by quadratures
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