A specialization inequality for tropical complexes

Cartwright, Dustin

doi:10.48550/arxiv.1511.00650

Cited by 1 publication

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“…Robustness in dimensions greater than 2 will play a critical role in the specialization theorem for tropical complexes [Car15b]. For Theorem 1.1 in this paper, the hypothesis on a degeneration is numerically faithful.…”

Section: C[[t]]mentioning

confidence: 86%

“…While Example 2.4 shows that the intersection matrix can have either 0 or 1 positive eigenvalues, we will be most interested in degenerations where every intersection matrix falls in the latter case. We want the weak tropical complex to reflect the algebraic geometry of the degeneration, and capturing the Hodge index theorem for surfaces is a key part of that geometry, as shown in Proposition 3.16 and in the follow-up papers [Car15a,Car15b].…”

Section: Degenerations and Their Tropical Complexesmentioning

confidence: 99%

“…The first, [Car15a], looks at the combinatorial properties of 2-dimensional tropical complexes, proving analogues of the Hodge index theorem and Noether's formula. The second, [Car15b], generalizes Baker's specialization inequality for degenerations of curves [Bak08] to degenerations of higher-dimensional varieties, using the definition of linear equivalence of divisors on tropical complexes given in this paper. Some of the developments in this paper are included to provide foundations for the those results.…”

Section: Introductionmentioning

confidence: 99%

“…While we have used the term "tropical complex" throughout the above discussion, many of the constructions and basic results in this paper work more generally for a more general class, called weak tropical complexes, and when possible, we state the results in those terms. Nonetheless, Theorem 1.1, as well as the main results of [Car15a,Car15b] work only for tropical complexes.…”

Section: Introductionmentioning

confidence: 99%

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Tropical complexes

Cartwright

2019

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We introduce tropical complexes, as an enrichment of the dual complex of a degeneration with additional data from non-transverse intersection numbers. We define cycles, divisors, and linear equivalence on tropical complexes, analogous both to the corresponding theories on algebraic varieties and to work on graphs and tropical curves. In addition, we establish conditions for the divisor-curve intersection numbers on a tropical curve to agree with the generic fiber of a degeneration. 'd especially like to thank Sam Payne for his many insightful suggestions and thoughtful comments on an early draft of this paper. I was supported by the National Science Foundation award number DMS-1103856 and National Security Agency award H98230-16-1-0019. I would also like to acknowledge the hospitality of the University of Georgia, where the first round of revisions to this paper were done. Degenerations and their tropical complexesThis section defines tropical complexes as combinatorial objects and to explains their construction from a degeneration. We begin by discussing the combinatorial properties of degenerations.We use degeneration to refer to a regular scheme X which is flat and proper over a discrete valuation ring R, and such that the special fiber X 0 is a reduced simple normal crossing divisor. In addition, we always assume that the residue field of the discrete valuation ring R is algebraically closed.

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Section: C[[t]]mentioning

confidence: 86%

Section: Degenerations and Their Tropical Complexesmentioning

confidence: 99%