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Penn 2014 Poster Session

 Saturday, November 1, 4:00-6:30pm,David Rittenhouse Laboratory, front lobby

Topology of the Complement of Certain Families of Trigonal Curves and Their Associated Dessins d’Enfants.  Mehmet Emin Aktas (Florida State)
I am reporting an on-going work concerning the topology of the complement of trigonal curves in the form
where and their associated Dessins d’Enfants. We improved a method to find braid monodromies of that type trigonal curves and by using Burau representations of braid groups, we could compute the Alexander polynomial of these curve complements. Also, we tried to classify the Dessin d’enfants of these curves and are still trying to compute the Alexander like invariants, such as the Alexander polynomial, other invariants based on representations of the braid group, etc. of a this type of trigonal curves in terms of their dessins.

A method for computing Segre classes of arbitrary projective varieties. Corey S Harris (Florida State)
We give an algorithm for computing Segre classes of subschemes of projective varieties.  The algorithm relies on computing a contribution of the subscheme to the degree of the ambient scheme (embedded in projective space) and comparing this with the results from taking successive hyperplane sections.  The output of these computations is a linear system of equations which determines the coefficients of the Segre class pushed forward to projective space. To our knowledge, this is the first algorithm to be able to compute Segre classes in ambient schemes other than projective space or toric varieties. One application is that the algorithm may be used to implement a routine for computing intersection products of projective varieties.

Prime congruences of idempotent semirings and a Nullstellensatz for
tropical polynomials.
Kalina Mincheva (Johns Hopkins)

 We present a new definition of prime congruences in additively idempotent semirings using twisted products. This class of congruences turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences it is shown that the intersection of all primes of a semiring
can be characterized by certain twisted power formulas. A complete description of prime congruences is given in the polynomial and Laurent polynomial semirings over the tropical semifield , the
semifield and the two element semifield . The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have
the same Newton polytope. It is then shown that every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension . An improvement of a recent result is
proven which can be regarded as a Nullstellensatz for tropical polynomials.

Martens theorem and the tropical Brill-Noether locus. Yoav Len (Yale)
We describe ongoing work on the geometry of the Brill-Noether locus of a tropical curve, and consequence for a tropical version of Martens theorem. The classical version of the theorem states that the dimension of the Brill–Noether locus equals exactly when the curve is hyperelliptic. We describe the relation between this inequality and the polyhedral structure of the tropical Brill-Noether locus. We present evidence that the theorem should hold tropically, and partial results towards a solution for the problem.

Kahler-Einstein metrics and birational geometry. Jesus Martinez-Garcia
(Johns Hopkins)
The existence of Kahler-Einstein edge metrics on where is a Fano manifold and an anticanonical cycle is equivalent to being log K-polystable. We use the dynamic alpha-invariant of to show the existence of these metrics for small angles when is any del Pezzo surface. This is particularly relevant for those which do not admit a Kahler-Einstein metric, where our edge metric gives an approximation of a Kahler-Einstein metric away from .

Berkovich skeletons for Hurwitz theory. Dhruv Ranganathan (Yale)
We explore the relationship between the geometry of the space of Hurwitz covers, its analytification, and the combinatorial geometry of its tropical analogue. The Berkovich analytification of the admissible cover compactification admits a strong deformation retract onto the skeleton, a finite polyhedral complex. We study the precise relationship between the skeleton and the moduli space of tropical admissible covers.. This allows for rigorous computations of algebraic Hurwitz numbers in terms of
"tropical" computations on the skeleton.

Artin fans in tropical geometry.  Martin Ulirsch (Brown)

Recent work by J. and N. Giansiracusa, myself, and O. Lorscheid suggests that the tropical geometry of a toric variety , or more generally of a logarithmic scheme , can be formalized as a "Berkovich analytification" of a scheme over the field with one element that is canonically associated to .

The goal of this poster is to introduce the theory of Artin fans, originally due to D. Abramovich and J. Wise, which can be used to lift rather unwieldy -geometric objects to the more familiar realm
of algebraic stacks. Artin fans are étale locally isomorphic to quotient stacks of toric varieties by their big tori and their glueing data has a completely combinatorial description in terms of Kato fans.

The poster is going to explain how to use the ideas surrounding the notion of  Artin fans to study tropicalization maps associated to toric varieties and logarithmic schemes. Surprisingly these techniques allow us to give a reinterpretation of Tevelev's theory of tropical compactifications that can
be generalized to compactifications of subvarieties in logarithmically smooth compactifcations of smooth varieties. For example, we can introduce definitions of tropical pairs and schoen varieties in terms of Artin fans that are equivalent to Tevelev's notions.

Non-abelian and Spherical Tropicalization. Tassos Vogiannou (UMass)

A tropical compactification of a subvariety of a torus is a compactification in a toric variety such that
   (i) is proper, and
   (ii) the structure morphism is faithfully flat.
The support of the fan corresponding to coincides with the tropicalization , which can be computed by other means; this suggests a way to compute . These compactifications possess  a set of nice properties. For instance if is smooth, any set of boundary divisors intersect in codimension (combinatorial normal crossings condition). Tropical compactifications were introduced and their existence was shown by Tevelev in 2007. We extend this idea to compactifications of subvarieties of spherical varieties for a non-abelian reductive group . We show their existence and provide some examples.

Moduli Spaces of Rational Maps. Lloyd West (CUNY)
I use the classical invariant theory of binary forms to construct the moduli space of degree d rational maps (i.e. morphisms of the projective line). As an example, I give an explicit description in the case . I also present an explicit solution to the question of field of moduli v.s. field of definition in this context