A moduli space of stable maps to the fibers of a fiber bundle is constructed. The new moduli space is a family version of the classical moduli space of stable maps to a non-singular complex projective variety. The virtual cycle for this moduli space is also constructed, and an analogue of Gromov-Witten invariants is defined. As an application, we recover the formula for the number of rational degree d curves in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1648_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {P}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>, whose image lies in a plane in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1648_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {P}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> (known as planar curves in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1648_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {P}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>), intersecting r general lines while passing through given s general points, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1648_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">$r + 2s = 3d + 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>+</mo> <mn>2</mn> <mi>s</mi> <mo>=</mo> <mn>3</mn> <mi>d</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, firstly proved by R. Mukherjee, R. Kumar Singh and the fourth named author.

Gromov-Witten invariants in family and Quantum cohomology

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2025年7月1日

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Indranil Biswas,
Nilkantha Das,
Jeongseok Oh,
Anantadulal Paul

摘要

A moduli space of stable maps to the fibers of a fiber bundle is constructed. The new moduli space is a family version of the classical moduli space of stable maps to a non-singular complex projective variety. The virtual cycle for this moduli space is also constructed, and an analogue of Gromov-Witten invariants is defined. As an application, we recover the formula for the number of rational degree d curves in $\mathbb {P}^3$ $P^{3}$ , whose image lies in a plane in $\mathbb {P}^3$ $P^{3}$ (known as planar curves in $\mathbb {P}^3$ $P^{3}$ ), intersecting r general lines while passing through given s general points, where $r + 2s = 3d + 2$ $r + 2 s = 3 d + 2$ , firstly proved by R. Mukherjee, R. Kumar Singh and the fourth named author.