Correspondence theorems for tropical curves I

Abstract

In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the obstruction cohomology class of degenerate algebraic curves. This enables us to judge whether a given degenerate curve can be deformed to a smooth curve or not in variety of situations. In this paper, we apply it to curves of genus one on degeneration of toric varieties. In particular, we obtain the necessary and sufficient condition for the realizability of tropical curves of genus one, extending various results obtained so far.

v2:Added references. v3:Corrected the case with higher edge weights, other modifications throughout. v4: Extensively reorganized, some parts are moved to other papers

References in corpus (2)

• Tropical Geometry and its applications
• Deformation theory via differential graded Lie algebras

Cited by in corpus (9)

• Brief introduction to tropical geometry
• Faithful realizability of tropical curves
• Tropical refined curve counting via motivic integration
• Lifting representations of finite reductive groups I: Semisimple conjugacy classes
• Harmonic tropical morphisms and approximation
• Enumerative geometry of elliptic curves on toric surfaces
• Tropical geometry and correspondence theorems via toric stacks
• Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants
• The Strominger-Yau-Zaslow conjecture and its impact