Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

By Vivek V. Shende. Compositio Mathematica, 2010

52 citations recorded by the source corpus.

Abstract Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande–Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar–Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a conjecture of Oblomkov and the present author identifies the Euler numbers of the Hilbert schemes with the ‘U(∞)’ invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/949095ab4c49ed0dd33e9d432af0ce6a7938695b

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In Discrete Mathematics and Combinatorics

1. I. Danilenko
2. I. Cherednik
3. Jacob Rasmussen
4. L. Gottsche
5. Alexei Oblomkov
6. Vivek V. Shende
7. E. Gorsky
8. M. Hedden

In Geometry and Topology

1. I. Danilenko
2. I. Cherednik
3. Vivek V. Shende
4. Jacob Rasmussen
5. Alexei Oblomkov
6. L. Gottsche
7. E. Gorsky
8. Peter S. Ozsv'ath

In Mathematical Physics

1. I. Cherednik
2. I. Danilenko
3. Vivek V. Shende
4. Alexei Oblomkov
5. Jacob Rasmussen
6. L. Gottsche
7. Peter S. Ozsv'ath
8. Zolt'an Szab'o

Citations in the corpus

• DAHA and iterated torus knots (2014, background): “i/2ZC due to [18] and [42], because these numbers are the multiplicities of C in the closures of the strata in the space of its versal deformation, stratified with respect to the geometric genus of the deformation curves.”
• Refined curve counting on complex surfaces (2012, methodology): “It was shown in [67] that the higher discriminants of the map C B ! B are precisely the loci x B , which by assumption agree with x B C .”
• The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, background): “cribed, we have ν(τg − τ′ g) 6= Γ , which is absurd. Definition 26. Fix a0,...,ak ∈ Γ. Use Σλ := Γ>a λ \ (a0,...,ak)Γ to index the set of indeterminates S:= {Sλ,i |i ∈ Σλ}. Let Va0,...,a k := Spec C[S], and define fλ ∈ O[S] by fλ := τa λ + X i∈Σλ τiSλ,i and form the ideal J := (f0,...,fk) ⊂ O[S]. C”