92 research outputs found
Welschinger invariants of toric Del Pezzo surfaces with non-standard real structures
The Welschinger invariants of real rational algebraic surfaces are natural
analogues of the Gromov-Witten invariants, and they estimate from below the
number of real rational curves passing through prescribed configurations of
points. We establish a tropical formula for the Welschinger invariants of four
toric Del Pezzo surfaces, equipped with a non-standard real structure. Such a
formula for real toric Del Pezzo surfaces with a standard real structure (i.e.,
naturally compatible with the toric structure) was established by Mikhalkin and
the author. As a consequence we prove that, for any real ample divisor on a
surfaces under consideration, through any generic configuration of
generic real points there passes a real rational curve
belonging to the linear system .Comment: An extended version of MPI Preprint no. MPIM2005-4
On refined count of rational tropical curves
We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts plane rational tropical curves with an unmarked vertex of arbitrary valency. It was motivated by the tropical enumeration of plane cuspidal tropical curves given by Y. Ganor and the author, which naturally led to consideration of plane tropical curves with an unmarked four-valent vertex. Another refined invariant counts rational tropical curves of a given degree in the Euclidean space of arbitrary dimension matching specific constraints, which make the spacial refined invariant similar to known planar invariants
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