[Remark: We refer to the two papers under review as first and second paper.]
Let \(S\) be a smooth projective rational surface and let \(D\) be an effective divisor on \(S\). Let \(V(D)\subset|D|\) be the closure of the locus of irreducible rational curves. If \(D\) has nonnegative self-intersection and \(V(D)\) is nonempty the dimension of \(V(D)\) is known: \[ r_0(D):=\dim V(D)=-(K_S\cdot D)-1. \] The problem that is studies here is to compute the degrees \(N(D):=\deg V(D)\) of these varieties as subvarities of \(|D|\cong\mathbb{P}^r\). Alternatively, \(N(D)\) is the number of irreducible rational curves in \(|D|\) that pass through \(r_0(D)\) general points of \(S\). If \(S\) is the projective plane \(\mathbb{P}^2\) and \(d=\deg D\), then one also uses the notation \(N(d)\) to denote the number of irreducible, rational curves of degree \(d\) passing through \(3d-1\) general points. If \(D\) and \(D'\) are effective divisors (or divisor classes) on a surface, we will say that \(D>D'\) if \(D-D'\) if effective and nonzero.
We will denote by \(\mathbb{F}_n\) the Hirzebruch surface \(\mathbb{F}_n=\mathbb{P}({\mathcal O}_{\mathbb{P}^1}\oplus{\mathcal O}_{\mathbb{P}^1}(n))\). On each \(\mathbb{F}_n\) with \(n\geq 1\) there exists a unique curve of negative self intersection, which we will denote by \(E\) and refer to as the exceptional curve on \(\mathbb{F}_n\). We will denote by \(F\) a fiber of the projection \(\mathbb{F}_n\to\mathbb{P}^1\); the classes of \(E\) and \(F\) generate the Picard group of \(\mathbb{F}_n\), with intersection pairing given by \[ E^2=-n;\quad (E\cdot F)=1\quad\text{and}\quad F^2=0. \] Another useful divisor class is the class of a complementary section, that is, a section \(C\) of the \(\mathbb{P}^1\)-bundle \(\mathbb{F}_n\to\mathbb{P}^1\) disjoint from \(E\). Since \((C\cdot E)=0\) and \((C\cdot F)=1\), we see that \(C\equiv E+nF\); so the classes \(C\) and \(F\) also generate the Picard group, with intersection numbers \[ C^2=n;\quad (C\cdot F)=1\quad\text{and}\quad F^2=0. \] For any positive integer \(m\), we will denote by \(V_m(D)\subset V(D)\subset|D|\) the closure of the locus of irreducible rational curves, \(X\) having contact of order at least \(m\) with \(E\) at a smooth point of \(X\). These varieties will also be referred to a Severi varieties. We set \(N_m(D):=\deg V_m(D)\) [see J. Harris, Invent. Math. 84, 445-461 (1986; Zbl 0596.14017)]. Until very recently, the basic enumerative problem of determining the degrees of Severi varieties was unsolved even in the case of \(\mathbb{P}^2\). In 1989, Ziv Ran described a recursive procedure for calculating the degrees of the Severi varieties parametrizing plane curves of any degree and genus. Recently, M. Kontsevich dicovered a beautiful and simple recursive formula in the case of rational curves on \(\mathbb{P}^2\). Kontsevich’s method was based on his description of a compactified space for maps of \(\mathbb{P}^1\) into the surface \(S=\mathbb{P}^2\); others were able to use the same method to derive similar formulas in the case of other surfaces \(S\) for which a Kontsevich-style moduli space existed, such as \(S=\mathbb{P}^1\times\mathbb{P}^1\), the ruled surface \(S=\mathbb{F}_1\) and del Pezzo surfaces.
The authors have the feeling that the reliance of Kontsevich’s method on the existence of a well-behaved moduli space was not essential. They are especially interested in whether a similar formula might be derived for the Hirzebruch surfaces \(\mathbb{F}_n\).
In the first paper, they succeed in recasting the Kontsevich method so as to remove the apparent dependence on the existence of a moduli space: As the authors set it up, it is necessary only to understand the degenerations of the rational curves in the one-parameter families corresponding to general one-dimensional linear sections of \(V(D)\). The resulting ‘cross-ratio method’ allowed to derive a complete recursion for all divisor classes on the ruled surface \(S=\mathbb{F}_2\) – that is, a formula expressing \(N(D)\) in terms of \(N(D')\) for \(D'<D\) – and a closed-form formula for certain divisor classes on the ruled surfaces \(\mathbb{F}_n\) for any \(n\). (In fact, compactifications of the moduli space of maps \(\mathbb{P}^1\to S\) do exist for these surfaces, but they contain in general many components, only one of which parametrizes generically irreducible rational curves and the others of which may have strictly larger dimension. Kontsevich’s method can be carried out in these cases, as was done by S. Kleiman and R. Piene; but the authors do not see how to use the resulting formulas to enumerate irreducible rational curves).
However, the authors are unable to go significantly beyond this point: A similarly derived formula in the first paper for the degrees \(N(D)\) of Severi varieties \(V(D)\) on \(\mathbb{F}_n\) expresses \(N(D)\) not solely in terms of \(N(D')\) for \(D'<D\), but also in terms of the degrees \(N_k(D'')\) of the Severi varieties \(V_k(D'')\) parametrizing curves with a point of \(k\)-fold tangency with a fixed curve \(E\subset S\). For example, if \(S=\mathbb{F}_3\), then \(N(D)\) is expressed as a function of \(N(D')\) and of \(N_2(D'')\), where \(N_2(D'')\) is the number of irreducible rational curves in \(|D''|\) that are simply tangent to \(E\) and pass through the appropriate number (that is, \(r_0(D'')-1)\) of general points of \(\mathbb{F}_3\). A complete recursion in this case would have required a similar analysis of linear sections of the Severi varieties \(V_2(D)\), which in turn would have necessitated an analysis of Severi varieties parametrizing curves with more complicated tangency conditions.
In the end, it seems that the authors need to deal with the degrees of these ‘tangential’ loci as well. This difficulty led them to the discovery of a computational technique different from and simpler than the cross-ratio method, which they describe in the second paper. It involves an analysis of the same basic object as the cross-ratio method – that is, the one-parameter family \(\chi\to\Gamma\) of rational curves through \(r_0(D)-1\) general points of \(S\) and their limits – but extracts more information from it. It is based on a description of the Néron-Severi group of a minimal desingularization of \(\chi\) (the authors refer to it as the ‘rational fibration method’). The main advantage of this technique for the present purposes is that the authors are in fact able to compute the degrees of the tangential loci involved; at least in all cases that they study. It also yields other related formulas, such as the number of irreducible rational curves having a node at a given general point \(p\in S\) and passing through \(r_0(D)-2\) other general points.