A quadratically enriched count of rational curves
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910035987136512 |
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| author | Kass, Jesse Leo Levine, Marc Solomon, Jake P. Wickelgren, Kirsten |
| author_facet | Kass, Jesse Leo Levine, Marc Solomon, Jake P. Wickelgren, Kirsten |
| contents | We define a quadratically enriched count of rational curves in a given divisor class passing through a collection of points on a del Pezzo surface $S$ of degree $\geq 3$ over a perfect field $k$ of characteristic $\neq 2,3.$ When $S$ is $\mathbb{A}^1$-connected, the count takes values in the Grothendieck-Witt group GW(k) of quadratic forms over $k$ and depends only on the divisor class and the fields of definition of the points. More generally, the count is a section of the Grothendieck-Witt sheaf evaluated on $π_0^{\mathbb{A}^1}$ of the restriction of scalars of $S$ corresponding to the fields of definition of the points. We also treat del Pezzo surfaces of degree $2$ under certain conditions. The curve count defined in the present work recovers Gromov-Witten invariants when $k = \mathbb{C}$ and Welschinger invariants when $k = \mathbb{R}.$
To obtain an invariant curve count, we define a quadratically enriched degree for an algebraic map $f$ of $n$-dimensional smooth schemes over a field $k$ under appropriate hypotheses. For example, $f$ can be proper, generically finite and oriented over the complement of a subscheme of codimension $2.$ This degree is compatible with F. Morel's GW(k)-valued degree of an $\mathbb{A}^1$-homotopy class of maps between spheres. For $k \subseteq \mathbb{C}$, this produces an enrichment of the topological degree of a map between manifolds of the same dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_01936 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A quadratically enriched count of rational curves Kass, Jesse Leo Levine, Marc Solomon, Jake P. Wickelgren, Kirsten Algebraic Geometry Algebraic Topology Symplectic Geometry Primary 14N35, 14F42, Secondary 53D45, 19G38 We define a quadratically enriched count of rational curves in a given divisor class passing through a collection of points on a del Pezzo surface $S$ of degree $\geq 3$ over a perfect field $k$ of characteristic $\neq 2,3.$ When $S$ is $\mathbb{A}^1$-connected, the count takes values in the Grothendieck-Witt group GW(k) of quadratic forms over $k$ and depends only on the divisor class and the fields of definition of the points. More generally, the count is a section of the Grothendieck-Witt sheaf evaluated on $π_0^{\mathbb{A}^1}$ of the restriction of scalars of $S$ corresponding to the fields of definition of the points. We also treat del Pezzo surfaces of degree $2$ under certain conditions. The curve count defined in the present work recovers Gromov-Witten invariants when $k = \mathbb{C}$ and Welschinger invariants when $k = \mathbb{R}.$ To obtain an invariant curve count, we define a quadratically enriched degree for an algebraic map $f$ of $n$-dimensional smooth schemes over a field $k$ under appropriate hypotheses. For example, $f$ can be proper, generically finite and oriented over the complement of a subscheme of codimension $2.$ This degree is compatible with F. Morel's GW(k)-valued degree of an $\mathbb{A}^1$-homotopy class of maps between spheres. For $k \subseteq \mathbb{C}$, this produces an enrichment of the topological degree of a map between manifolds of the same dimension. |
| title | A quadratically enriched count of rational curves |
| topic | Algebraic Geometry Algebraic Topology Symplectic Geometry Primary 14N35, 14F42, Secondary 53D45, 19G38 |
| url | https://arxiv.org/abs/2307.01936 |