A specialization inequality for tropical complexes

Abstract

We prove a specialization inequality relating the dimension of the complete linear series on a variety to the tropical complex of a regular semistable degeneration. Our result extends Baker's specialization inequality to arbitrary dimension.

Tropical theta functions and Riemann–Roch inequality for tropical Abelian surfaces

“…In this section, we show that h 0 (X, D) introduced in Cartwright [6] is equal to the topological dimension of Γ(X, O(D)).…”

“…Cartwright conjectured a Riemann-Roch inequality for weak tropical surfaces in [6,Conjecture 3.6]. He introduced a higher dimensional analogue h 0 (X, D) of r(D) + 1 and proposed a Riemann-Roch inequality by omitting h 1 (X, D) and assuming the Serre duality i.e.…”

“…Let X be a tropical torus and let D be a divisor on X. Then h 0 (X, D) defined by Cartwright [6] coincides with the dimension of Γ(X, O(D)) as a convex polyhedron.…”

Riemann-Roch inequality for smooth tropical toric surfaces

“…A higher dimensional generalization of r(D) (precisely, r(D) + 1) is introduced in Cartwright [6], denoted by h 0 (X, D), and formulate a Riemann-Roch inequality for tropical surfaces. The author computed the value of h 0 (X, D) for a divisor D on a tropical torus X and showed that the Riemann-Roch inequality formulated by Cartwright holds for tropical Abelian surfaces in [21].…”