A Fock space approach to Severi degrees of Hirzebruch surfaces

The classical Severi degree counts the number of algebraic curves of fixed genus and class passing through some general points in a surface. In this paper we study Severi degrees as well as several types of Gromov–Witten invariants of the Hirzebruch surfaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript k"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the relationship between these numbers. To each Hirzebruch surface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript k"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we associate an operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper M Subscript upper F Sub Subscript k Baseline element-of script upper H left-bracket double-struck upper P Superscript 1 Baseline right-bracket"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">M</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {M}_{F_k} \in \mathcal {H}[\mathbb {P}^1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acting on the Fock space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F left-bracket double-struck upper P Superscript 1 Baseline right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {F}[\mathbb {P}^1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Generating functions for each of the curve-counting theories we study here on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript k"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be expressed in terms of the exponential of the single operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper M Subscript upper F Sub Subscript k"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">M</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {M}_{F_k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and counts on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be expressed in terms of the exponential of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper M Subscript upper F 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">M</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {M}_{F_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Several previous results can be recovered in this framework, including the recursion of Caporaso and Harris [Invent. Math. 131 (1998), pp. 345–392] for enumerative curve counting on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the generalization by Caporaso and Harris and later Vakil [Manuscripta Math. 102 (2000), pp. 53–84] to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript k"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the relationship of Abramovich–Bertram [<italic>The formula</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="12 equals 10 plus 2 times 1"> <mml:semantics> <mml:mrow> <mml:mn>12</mml:mn> <mml:mo>=</mml:mo> <mml:mn>10</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">12 = 10 + 2 \times 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>and its generalizations: counting rational curves on</italic> <bold>F</bold>2, Amer. Math. Soc., Providence, RI, 2001, pp. 83–88] between the enumerative curve counts on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 0"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 2"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove an analog of Abramovich–Bertram for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 1"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 3"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also obtain two differential equations satisfied by generating functions of relative Gromov–Witten invariants on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript k"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One of these recovers the differential equation of Getzler [J. Amer. Math. Soc. 10 (1997), pp. 973–998] and Vakil [Manuscripta Math. 102 (2000), pp. 53–84].