arXiv subjects
Publications and source records attributed to Franziska Schroeter.
We propose a geometric interpretation of Block and Göttsche's refined tropical curve counting invariants in terms of virtual $χ_{-y}$-specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that this interpretation is correct for linear series of genus 1, and in arbitrary genus after specializing from $χ_{-y}$ to Euler characteristic.
F. Block and L. Göttsche introduced refined tropical invariants of toric surfaces that intertwine tropical Gromov-Witten and Welschinger invariants of toric surfaces. L. Göttsche and the first author introduced refined broccoli invariants that intertwine some genus zero descendant tropical invariants and broccoli invariants of toric surfaces. In this note, we extend the refined broccoli invariants to the genus one case.
We introduce a tropical enumerative invariant depending on a variable y which generalizes the tropical refined Severi degree. We show that this refined broccoli invariant is indeed independent of the point configuration, and that it specializes to a tropical descendant Gromov-Witten invariant for y=1 and to the corresponding broccoli invariant for y=-1. Furthermore, we define tropical refined descendant Gromov-Witten invariants which equal the corresponding refined broccoli invariants giving a new insight to the nature of broccoli invariants. We discuss various possible generalizations, e.g. to refinements of bridge curves and Welschinger curves.
The computation of amoebas has been a challenging open problem for the last dozen years. The most natural approach, namely to compute an amoeba via its boundary, has not been practical so far since only a superset of the boundary, the contour, is understood in theory and computable in practice. We define and characterize the extended boundary of an amoeba, which is sensitive to some degenerations that the topological boundary does not detect. Our description of the extended boundary also allows us to distinguish between the contour and the boundary. This gives rise not only to new structural results in amoeba theory, but in particular allows us to compute hypersurface amoebas via their boundary in any dimension. In dimension two this can be done using Gröbner bases alone. We introduce the concept of amoeba bases, which are sufficient for understanding the amoeba of an ideal. We show that our characterization of the boundary is essential for the computation of these amoeba bases and we illustrate the potential of this concept by constructing amoeba bases for linear systems of equations.
In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points - for arbitrary choices of the directions of the ends of the curves, possibly with higher weights, and also if some of the ends are fixed. In the toric Del Pezzo case we show that these broccoli invariants are equal to the Welschinger invariants (with real and complex conjugate point conditions), thus providing a proof of the independence of Welschinger invariants of the point conditions within tropical geometry. The general case gives rise to a tropical Caporaso-Harris formula for broccoli curves which suffices to compute all Welschinger invariants of the plane.
In the first part of this paper, we discuss the notion of irreducibility of cycles in the moduli spaces of n-marked rational tropical curves. We prove that Psi-classes and vital divisors are irreducible, and that locally irreducible divisors are also globally irreducible for n \leq 6. In the second part of the paper, we show that the locus of point configurations in (\R^2)^n in special position for counting rational plane curves (defined in two different ways) can be given the structure a tropical cycle of codimension 1. In addition, we compute explicitly the weights of this cycle.