Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An invitation to the enumerative geometry of degenerations

Dhruv Ranganathan

Abstract

This expository article is an introduction to logarithmic Gromov--Witten (GW) theory. We discuss how to study the GW theory of a smooth projective variety via simple normal crossings degenerations. We survey several approaches to constructing well-behaved, virtually smooth moduli spaces of stable maps to such degenerations. Each irreducible component of the special fiber of a degeneration determines a pair consisting of a variety and a normal crossings divisor, and these pairs carry their own logarithmic GW theory. We explain how the GW theory of the general fiber can be expressed in terms of the logarithmic GW theory of these pairs. Finally, we discuss applications to tautological classes on the moduli space of curves.

An invitation to the enumerative geometry of degenerations

Abstract

This expository article is an introduction to logarithmic Gromov--Witten (GW) theory. We discuss how to study the GW theory of a smooth projective variety via simple normal crossings degenerations. We survey several approaches to constructing well-behaved, virtually smooth moduli spaces of stable maps to such degenerations. Each irreducible component of the special fiber of a degeneration determines a pair consisting of a variety and a normal crossings divisor, and these pairs carry their own logarithmic GW theory. We explain how the GW theory of the general fiber can be expressed in terms of the logarithmic GW theory of these pairs. Finally, we discuss applications to tautological classes on the moduli space of curves.
Paper Structure (50 sections, 17 theorems, 127 equations, 10 figures)

This paper contains 50 sections, 17 theorems, 127 equations, 10 figures.

Table of Contents

  1. Introduction
  2. A motivating result
  3. Three things about GW theory
  4. Deformations, degenerations, semistable reduction
  5. Outline
  6. The degeneration theory
  7. The strata theory
  8. The rest of the document
  9. Further reading
  10. Curves in degenerations
  11. Two basic requirements
  12. A cheap trick
  13. Expansions
  14. Predeformable stable maps
  15. The stack of expansions
  16. ...and 35 more sections

Key Result

Theorem 1

For all values of $g$ and $d$, the $0$-cycle $\mathfrak m_g(X_5,d)$ is proportional to the top Chern class of the tangent bundle of $\overline{\mathsf M}\newline_g$ in the Chow group of $0$-cycles on $\overline{\mathsf M}\newline_g$. Equivalently, the class $\mathfrak m_g(X_5,d)$ is proportional to

Figures (10)

  • Figure 1: A cartoon of a surface degenerating into three components meeting transversely.
  • Figure 2: Starting with three surfaces meeting at a point, visualized on the right, blowup at the triple point and then each double curve, and base change. The result is an expansion that might be visualized as on the left.
  • Figure 3: The curve on the left is dimensionally transverse, while the two on the right are not.
  • Figure 4: A non-predeformable map.
  • Figure 5: The picture at the level of fans of a smoothing predeformable map.
  • ...and 5 more figures

Theorems & Definitions (40)

  • Theorem 1
  • Definition 2: Cheap logarithmic maps
  • Theorem 3
  • proof
  • Definition 4
  • Theorem 5
  • proof
  • Theorem 6: Abramovich's land of fairytales
  • proof
  • Theorem 7: Log maps factor through expansions
  • ...and 30 more