Volume 13, issue 4 (2009)

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The $3$–fold vertex via stable pairs

Rahul Pandharipande and Richard P Thomas

Geometry & Topology 13 (2009) 1835–1876
Abstract

The theory of stable pairs in the derived category yields an enumerative geometry of curves in $3$–folds. We evaluate the equivariant vertex for stable pairs on toric $3$–folds in terms of weighted box counting. In the toric Calabi–Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities.

The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent theory for stable pairs.

Keywords
curve, threefold, Gromov–Witten, toric
Mathematical Subject Classification 2000
Primary: 14N35
Secondary: 14M25, 14D20, 14J30