We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.Comment: 36 pages, final version to appear in Duk

Abstract. In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima.

We prove that the Chow quotient parameterizing configurations of n points in P d which generically lie on a rational normal curve is isomorphic to M 0,n , generalizing the well-known d = 1 result of Kapranov. In particular, M 0,n admits birational morphisms to all the corresponding geometric invariant theory (GIT) quotients. For symmetric linearizations, the polarization on each GIT quotient pulls back to a divisor that spans the same extremal ray in the symmetric nef cone of M 0,n as a conformal blocks line bundle. A symmetry in conformal blocks implies a duality of point-configurations that comes from Gale duality and generalizes a result of Goppa in algebraic coding theory. In a suitable sense, M 0,2m is fixed pointwise by the Gale transform when d = m − 1 so stable curves correspond to self-associated configurations.Licensed to Univ of Calif, Santa Barbara. Prepared on Sun Jul 5 01:52:54 EDT 2015 for download from IP 128.111.121.42.License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf CONFORMAL BLOCKS AND RATIONAL NORMAL CURVES 775GIT quotient V d,n / / L SL d+1 comes with a polarization, which by Theorem 1.1 can be pulled back to a line bundle on M 0,n . For the S n -invariant linearization L, denote the GIT polarization by L d ∈ Pic(V d,n / / L SL d+1 ).Theorem 1.2. The line bundles ϕ * L d and D sl n d+1 span the same ray in N 1 (M 0,n ).By [Fak11, Remark 5.3], the line bundle D sl n 2 can be identified with the sl 2 conformal blocks determinant inducing the map M 0,n → (P 1 ) n / / L SL 2 , so the d = 1 case of this theorem is due to Fakhruddin. Corollary 1.3. For d = 1, . . . , n 2 −1, the line bundles ϕ * L d span distinct extremal rays of the symmetric nef cone of M 0,n .In [AGSS10] it was shown that D sl n 2 , . . . , D sl n n 2