1 S-polynomials eliminating the leading term Buchberger's criterion and algorithm 2 Wavelet Design construct wavelet filters 3 Proof of the Buchberger Criterion two lemmas proof of the Buchberger criterion termination and elimination

Gröbner Bases와 응용

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: 
- the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold 
- the diagonal coinvariant algebra on r sets of n variables 
- the cohomology and tautological ring of the moduli space of n-pointed curves 
- the space of polynomials on rank varieties of n x n matrices 
- the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. 
The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.

FI-modules and stability for representations of symmetric groups

© Foundation Compositio Mathematica, 1972, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

An analytic construction of degenerating curves over complete local rings ; An analytic construction of degenerating abelian varieties over complete rings

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.

/pdf/equations-of-tropical-varieties-2ah40xv7op.pdf

Equations of tropical varieties

In this work, we argue that the $\alpha'\to 0$ limit of closed string theory scattering amplitudes is a tropical limit The motivation is to develop a technology to systematize the extraction of Feynman graphs from string theory amplitudes at higher genus An important technical input from tropical geometry is the use of tropical theta functions with characteristics to rigorously derive the worldline limit of the worldsheet propagator This enables us to perform a non-trivial computation at two loops: we derive the tropical form of the integrand of the genus-two four-graviton type II string amplitude, which matches the direct field theory computations At the mathematical level, this limit is an implementation of the correspondence between the moduli space of Riemann surfaces and the tropical moduli space

Tropical Amplitudes

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.

Tropicalization of Classical Moduli Spaces

We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.

/pdf/bitangents-of-tropical-plane-quartic-curves-angjj82l8x.pdf

Bitangents of tropical plane quartic curves

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schl\"afli graphs, respectively. There are 27 trees on each tropical cubic surface, attached to a bounded complex with up to 73 polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic types of such surfaces.

Tropicalization of Del Pezzo Surfaces

The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in . We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Gopel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble's quartic hypersurfaces using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus-3 moduli spaces appear alongside toric and tropical methods.

The Universal Kummer Threefold

The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in 7-dimensional projective space. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Gopel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble's quartic hypersurface using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus 3 moduli spaces appear alongside toric and tropical methods.

Qingchun Ren

Papers

Tropicalization of Classical Moduli Spaces

Bitangents of tropical plane quartic curves

Tropicalization of Del Pezzo Surfaces

The Universal Kummer Threefold

The Universal Kummer Threefold