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The chi-y genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces
Algebraic Geometry · Submitted 16 Jul 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We give an explicit expression in quasi-Jacobi forms for the chi-y genus of the restriction of the Hilbert scheme to a general linear subsystem.
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The abstract, sentence by sentence
For an ample line bundle on an Abelian or K3 surface, minimal with respect to the polarization, the relative Hilbert scheme of points on the complete linear system is known to be smooth. We give an explicit expression in quasi-Jacobi forms for the chi-y genus of the restriction of the Hilbert scheme to a general linear subsystem. This generalizes a result of Yoshioka and Kawai for the complete linear system on the K3 surface, a result of Maulik, Pandharipande, and Thomas on the Euler characteristics of linear subsystems on the K3 surface, and a conjecture of the authors.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For an ample line bundle on an Abelian or K3 surface, minimal with respect to the polarization, the relative Hilbert scheme of points on the complete linear system is known to be smooth. | background 1.00 | 0.04 | 0.05 | asserted 1.00 |
| s1 | We give an explicit expression in quasi-Jacobi forms for the chi-y genus of the restriction of the Hilbert scheme to a general linear subsystem. | contribution 0.60 | 0.96 | 0.81 | asserted 1.00 |
| s2 | This generalizes a result of Yoshioka and Kawai for the complete linear system on the K3 surface, a result of Maulik, Pandharipande, and Thomas on the Euler characteristics of linear subsystems on the K3 surface, and a conjecture of the authors. | interpretation 0.67 | 0.45 | 0.20 | asserted 0.99 |
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The paper
Original abstract
For an ample line bundle on an Abelian or K3 surface, minimal with respect to the polarization, the relative Hilbert scheme of points on the complete linear system is known to be smooth. We give an explicit expression in quasi-Jacobi forms for the chi-y genus of the restriction of the Hilbert scheme to a general linear subsystem. This generalizes a result of Yoshioka and Kawai for the complete linear system on the K3 surface, a result of Maulik, Pandharipande, and Thomas on the Euler characteristics of linear subsystems on the K3 surface, and a conjecture of the authors.
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How the literature uses it
5 arXiv papers cite this in 7 sentences
Papers citing it · 3 sentences without a confident label
Compares with it 1 paper
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Writing $DG_2(y,q)=F_0(y,q)/y-2+y^-1$ we have [equation] A similar computation has been done in [ref] .
Refined node polynomials via long edge graphs · Counting curves with prescribed multiple points · 0.92
Cites it as support 1 paper
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We have the following evidence for this conjecture: In [ref] Gconj is proven for $S$ an abelian or K3 surface, and the positivity of $^(S,L), δ(y)$ follows for all line bundles $L$ on $S$.
Refined curve counting with tropical geometry · Refined node polynomials · 0.99
Cites it as background 2 papers
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The pairs spaces were introduced to study enumerative geometry on 3-folds [ref] ; but more relevant to our present work on locally planar curves are their uses in studying curves on surfaces [ref] , knot invariants [ref] , and the geometry of the Hitchin system [ref] .
A support theorem for Hilbert schemes of planar curves, II · Introduction · 0.92
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Tropical Welschinger invariants are also related to refined Severi degrees [ref] .
Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces · Relation with tropical Welschinger invariants and refined Severi degrees · 0.91
What it cites · 24 sentences without a confident label
Uses the method or tool of 2 papers
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As explained in [ref] , for a linear subsystem $^δ⊂|L|$ such that the relative Hilbert schemes $^[n]_[S,L],δ →^δ$ are all smooth – e.g., a general $δ$-dimensional linear subsystem when $L$ is $δ$-very-ample [ref] – we may extract the $χ_-y$ genera by taking a residue:
Testing Ultrafast Two-Photon Spectral Amplitudes via Optical Fibres · Universality arguments · 0.89
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As explained in [ref] , for a linear subsystem $^δ⊂|L|$ such that the relative Hilbert schemes $^[n]_[S,L],δ →^δ$ are all smooth – e.g., a general $δ$-dimensional linear subsystem when $L$ is $δ$-very-ample [ref] – we may extract the $χ_-y$ genera by taking a residue:
Refined curve counting on complex surfaces · Universality arguments · 0.89
Compares with 1 paper
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Then we define[ Note this differs from [ref] by the normalization by $y^-/2$. ] [equation]
Refined curve counting on complex surfaces · Universality arguments · 0.88
Cites as support 1 paper
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In the good situation where $[S, L]$ comes from a line bundle on a surface with no higher cohomology and the appropriate relative Hilbert schemes are nonsingular, $N^δ|_y=1$ counts the number of $δ$ nodal curves in a general $^δ⊂|L|$ by [ref] .
Testing Ultrafast Two-Photon Spectral Amplitudes via Optical Fibres · Introduction · 0.97
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