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UNIVERSAL FAMILIES OF RATIONAL TROPICAL CURVES
, 2011
"... We introduce the notion of families of nmarked smooth rational tropical curves over smooth tropical varieties and establish a onetoone correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of nmarked abstract rational t ..."
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We introduce the notion of families of nmarked smooth rational tropical curves over smooth tropical varieties and establish a onetoone correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of nmarked abstract rational tropical curves Mn.
Correspondence Theorems via Tropicalizations of Moduli Spaces
, 2014
"... We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that ..."
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We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind we prove a general correspondence theorem for enumerative problems which are defined via "evaluation maps " in both the algebraic and tropical world. Applying this to our motivational example we show that the tropicalizations of the curves in a given toric variety which intersect the boundary divisors in their interior and with prescribed multiplicities, and pass through an appropriate number of generic points are precisely the tropical curves in the corresponding tropical toric variety satisfying the analogous condition. Moreover, the intersectiontheoretically defined multiplicities of the tropical curves are equal to the numbers of algebraic curves tropicalizing to them.
COMBINATORICS OF TROPICAL HURWITZ CYCLES
, 2014
"... We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of such points the resulting cycles are weakly irreducible, ..."
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We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of such points the resulting cycles are weakly irreducible, i.e. an integer multiple of an irreducible cycle. We study how Hurwitz cycles can be written as divisors of rational functions and show that they are numerically equivalent to a tropical version of a representation as a sum of boundary divisors. The results and counterexamples in this paper were obtained with the help of atint, an extension for polymake for tropical intersection theory.