The nef cones of and minimal-degree curves in the Hilbert schemes of points on certain surfaces (Q312766)

Let \(X\) be a smooth projective complex surface. Long ago \textit{J. Fogarty} showed that the Hilbert scheme \(X^{[n]}\) parametrizing schemes of finite length \(n\) is smooth and irreducible of dimension \(2n\) [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. Here the authors describe the nef cone of \(X^{[n]}\) and the dual cone \(\mathrm{NE} (X^{[n]})\) spanned by effective curves when \(n \geq 2\) in terms of the corresponding cones on \(X\) under certain conditions.NEWLINENEWLINETo state the main theorem, suppose that \(H^1(X, {\mathcal O}_X)=0\). If \(B_n \subset X^{[n]}\) denotes the boundary divisor consisting of non-reduced subschemes \(Z \subset X\), there is an isomorphism \(\Psi: \mathrm{Pic} (X^{n]}) \cong \mathrm{Pic} (X) \oplus \mathbb Z \cdot (B_n / 2)\), due to \textit{J. Fogarty} [Am. J. Math. 95, 660--687 (1973; Zbl 0299.14020)]. Now assume that (a) the nef cone of \(X\) is spanned by \(F_1, \dots, F_t\) and \(\mathrm{NE} (X)\) is spanned by curves \(C_1, \dots, C_t\) satisfying \(F_i \cdot C_j = \delta_{i,j}\) and (b) the divisor \(L=(n-1) \sum F_i\) is \((n-1)\)-very ample in the sense of \textit{M. Beltrametti} and \textit{A. J. Sommese} [in: Problems in the theory of surfaces and their classification. Papers from the meeting held at the Scuola Normale Superiore, Cortona, Italy, October 10-15, 1988. London: Academic Press; Rome: Istituto Nazionale di Alta Matematica Francesco Severi. 33--48; appendix: 44--48 (1991; Zbl 0827.14029)], meaning that \(H^0(X,L) \to H^0(X, {\mathcal O}_Z \otimes L)\) is surjective for each \(Z \in X^{[n]}\). Then the authors prove that (1) the nef cone of \(X^{[n]}\) is spanned by \(D_{F_1}, \dots, D_{F_t}, (n-1) \sum D_{F_i} - B_n / 2\), where \(D_{F_i} \in \mathrm{Pic} (X^{[n]}\) corresponds to \(F_i\) under the isomorphism \(\psi\) and (2) \(\mathrm{NE} (X^{[n]})\) is spanned by the classes NEWLINE\[NEWLINE \beta_{C_1} - (n-1) \beta_n, \dots, \beta_{C_t} - (n-1) \beta_n, \beta_n, NEWLINE\]NEWLINE where \(\beta_{C_i} = \{x + x_1 + \dots + x_{n-1}: x \in C_i\}\) for some fixed points \(x_1, \dots, x_{n-1}\) not on the \(C_i\) and \(\beta_n = \{ Z + x_2 + \dots + x_{n-1} \in X^{[n]}: \mathrm{Supp} (Z) = \{x_1\} \}\). This extends earlier work of \textit{W.-P. Li} et al. [Contemp. Math. 322, 89--96 (2003; Zbl 1057.14012)].NEWLINENEWLINEThey apply the theorem to a Hirzebruch surface \(X\), recovering a result of \textit{A. Bertram} and \textit{I. Coskun} [in: Birational geometry, rational curves, and arithmetic. Based on the symposium ``Geometry over closed fields'', St. John, UK, February 2012. New York, NY: Springer. 15--55 (2013; Zbl 1273.14032)]. They also classify the curves whose homology classes are in the list above which have minimal degree in the sense that their intersection numbers with certain very ample divisors are all equal to one, proving that their moduli spaces are smooth of the expected dimension.