The formula \(12=10+2\times 1\) and its generalizations: Counting rational curves on \(F_2\) (Q2752189)

The authors use the deformation invariance property of Gromov-Witten invariants to give a recursive formula for the rational enumerative invariants of the Hirzebruch surface \(\mathbb{F}_2\).NEWLINENEWLINEIt is known that there exist a one parameter deformation \(X\to S\) having \(F_2\) as a special fiber and \({\mathbb P}^1\times {\mathbb P}^1\) as a generic fiber. There is a ruling on the family \(X\to S\) which induces the ruling of \(\mathbb{F}_2\) and one of the two rulings in \({\mathbb P}^1\times {\mathbb P}^1\); denote the fiber of this ruling by \(F\). The Picard group of \(F_2\) is generated by the class of \(F\) and by the class of a generic section, denoted \(C_2\). The Picard group of \({\mathbb P}^1\times {\mathbb P}^1\) is generated by the rulings. If we denote the fiber of the ruling transversal to \(F\) by \(C_0\), then the class \(C_2\) in \(F_2\) deforms to \(C_0+F\) in \({\mathbb P}^1\times {\mathbb P}^1\). This implies that, for any fixed non-negative integers \(a\) and \(b\), the divisor class \(aC_2+bF\) on \(F_2\) deforms to the class \(aC_0+(a+b)F\) on \({\mathbb P}^1\times {\mathbb P}^1\), and so there is the following equation between Gromov-Witten invariants: NEWLINE\[NEWLINEN_{\text{F}_2}(aC_2+bF)=N_{{\mathbb P}^1\times {\mathbb P}^1}(aC_0+(a+b)F).NEWLINE\]NEWLINE Genus zero Gromov-Witten invariants of \({\mathbb P}^1\times {\mathbb P}^1\) are enumerative, so \(N_{{\mathbb P}^1\times {\mathbb P}^1}(aC_0+(a+b)F)\) equals the number of rational curves in \({\mathbb P}^1\times {\mathbb P}^1\) of class \(aC_0+(a+b)F\) and passing through \(4a+2b-1\) general points. The corresponding enumerative invariant of \(F_2\) is \(N_{F_2,\text{enum.}}(aC_2+bF)\) which counts the number of irreducible rational nodal curves of class \(aC_2+bF\) in \(F_2\) passing through \(4a+2b-1\) general points.NEWLINENEWLINEIn this paper, the authors describe the relation between the two invariants \(N_{F_2,\text{enum.}}(aC_2+bF)\) and \(N_{\text{F}_2}(aC_2+bF)\) via the deformation space \(X\to S\) and find an explicit formula relating them: NEWLINE\[NEWLINE N_{\text{F}_2}(aC_2+bF)=\sum_{k=0}^{a-1}\binom{b+2k}{k} N_{F_2,\text{enum.}}\bigl((a-k)C_2+(b+2k)F\bigr) NEWLINE\]NEWLINE This formula is different from that of \textit{L.~Caporaso} and \textit{J.~Harris} [Compos. Math. 113, 155--208 (1998; Zbl 0930.14036)], but the resulting numbers coincide. Moreover, if one takes \((a,b)=(2,0)\), one finds \(12=10+2\times1\), whence the title of this paper.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00024].