Abstract:Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and covers of tropical curves maps between metric graphs satisfying certain conditions. In this short survey, we offer an introduction to the combinatorial theory of abstract tropical curves and covers of curves, and their moduli spaces, and we showcase three results demonstrating how this theory can be applied in algebraic geometry.
“…In Figures 4 and 5 , the (possibly folded) positive orthants are glued with respect to degeneration of the corresponding tree types, which is shown in Figure 6. We refer to [16 , Section 2] and [ 5, Section 4] for more details. Figure 6. Degenerations of trees.…”
Section: Tropical Invariants For General Group Actionsmentioning
In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification
$\overline{M}_{0,n}$
.
“…In Figures 4 and 5 , the (possibly folded) positive orthants are glued with respect to degeneration of the corresponding tree types, which is shown in Figure 6. We refer to [16 , Section 2] and [ 5, Section 4] for more details. Figure 6. Degenerations of trees.…”
Section: Tropical Invariants For General Group Actionsmentioning
In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification
$\overline{M}_{0,n}$
.
We prove a q-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov–Witten invariants with a $$\lambda _g$$
λ
g
class in toric surfaces. Specifically, a generating series of such logarithmic Gromov–Witten invariants agrees with a q-refined count of rational tropical curves satisfying higher valency conditions. As a corollary, we obtain a geometric proof of the deformation invariance of this tropical count. In particular, our results give an algebro-geometric meaning to the tropical count defined by Blechman and Shustin. Our strategy is to use the logarithmic degeneration formula, and the key new technique is to reduce to computing integrals against double ramification cycles and connect these integrals to the non-commutative KdV hierarchy.
A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly within the framework of progress Maddy offers, is developed with a special focus on mathematicians’ peer-review practice.
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