arXiv subjects

Thomas Blomme

Publications and source records attributed to Thomas Blomme.

21 records · Page 2Linked to original sources

arXiv2020

Computation of refined toric invariants II

In 2015, G.~Mikhalkin introduced a refined count for real rational curves in toric surfaces. The counted curves have to pass through some real and complex points located on the toric boundary of the surface, and the count is refined according to the value of a so called quantum index. This count happens only to depend on the number of complex points on each toric divisors, leading to an invariant. First, we give a way to compute the quantum index of any oriented real rational curve, getting rid of the previously needed "purely imaginary" assumption on the complex points. Then, we use the tropical geometry approach to relate these classical refined invariants to tropical refined invariants, defined using Block-Göttsche multiplicity. This generalizes the result of Mikhalkin relating both invariants in the case where all the points are real, and the result of the author where complex points are located on a single toric divisor.

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arXiv2020

A Tropical Computation of Refined Toric Invariants

In arXiv:1505.04338(4), G. Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through certain conjugation invariant set of points on the toric boundary of the surface. Such a set consists of real points and pairs of complex conjugated points. He then proved that the result of this refined count depends only on the number of pairs of complex conjugated points on each toric divisor. Using the tropical geometry approach and the correspondence theorem, he managed to compute these invariants if all the points of the configuration are real. In this paper we address the case when the configuration contains some pairs of conjugated points, all belonging to the same component of the toric boundary.

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arXiv2019

A Caporaso-Harris type Formula for relative refined invariants

G. Mikhalkin introduced a refined count for real rational curves in a toric surface which pass through some points on the toric boundary of the surface. The refinement is provided by the value of a so-called quantum index. Moreover, he proved that the result of this refined count does not depend on the choice of the points. The correspondence theorem allows one to compute these invariants using the tropical geometry approach and the refined Block-Göttsche multiplicities. In this paper we give a recursive formula for these invariants, that leads to an algorithm to compute them.

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