Conformal blocks and rational normal curves (Q2857297)

This paper generalizes \textit{M. M. Kapranov}'s [Adv. Sov. Math. 16(2), 29--110 (1993; Zbl 0811.14043)] well-known construction of \(\overline{\mathcal{M}}_{0,n}\) as the Chow quotient \((\mathbb{P}^1)^d /\!/_{\text{Ch}} \text{SL}_2\) by considering the set NEWLINE\[NEWLINE U_{d,n} := \left\{ (p_1, \dots, p_n) \in (\mathbb{P}^d)^n \,\mid\, p_i \text{ are distinct points lying on a rational normal curve} \right\} NEWLINE\]NEWLINE and its closure \(V_{d,n} := \overline{U_{d,n}}\) in \((\mathbb{P}^d)^n\). This closure is shown to consist of all configurations of (possibly coinciding) points that lie on a degeneration of a rational normal curve. The author shows that for \(d \leq n-3\) the Chow quotient \(V_{d,n} /\!/_{\text{Ch}} \text{SL}_{d+1}\) is isomorphic to \(\overline{\mathcal{M}}_{0,n}\), and that for any effective linearization \(L\) there exists a morphism to the GIT quotient \(\varphi: \overline{\mathcal{M}}_{0,n} \to V_{d,n} /\!/_L \text{SL}_{d+1}\), extending the isomorphism \(\mathcal{M}_{0,n} \widetilde{\to}\; U_{d,n} / \text{SL}_{d+1}\). Every effective linearization can be encoded by a tuple \(L = (x_1, \dots, x_n)\) of rational numbers with the conditions that \(0 \leq x_i \leq 1\) and \(\sum_{i=1}^n x_i = d+1\), and the morphism \(\varphi\) is shown to factor through the corresponding moduli space \(\overline{\mathcal{M}}_{0,L}\) of \(L\)-weighted pointed stable curves that was constructed in [\textit{B. Hassett}, Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)].NEWLINENEWLINEMoreover it is shown that if \(L = (\frac{d+1}{n}, \dots, \frac{d+1}{n})\) is the unique \(S_n\)-invariant linearization and \(\mathcal{L}_d\) denotes the GIT polarization on \(V_{d,n} /\!/_L \text{SL}_{d+1}\), then \(\varphi^* \mathcal{L}_d\) spans the same ray in \(N^1(\overline{\mathcal{M}}_{0,n})\) as a certain conformal blocks divisor studied previously in [\textit{M. Arap} et al., Int. Math. Res. Not. 2012, No. 7, 1634--1680 (2012; Zbl 1271.14034)]. Results from that paper imply that for \(d = 1, \dots, \lfloor \frac{n}{2} \rfloor - 1\) the line bundles \(\varphi^* \mathcal{L}_d\) span distinct extremal rays of the symmetric nef cone of \(\overline{\mathcal{M}}_{0,n}\). On the other hand, using the classical Gale transform the author shows that \(\varphi^* \mathcal{L}_d\) and \(\varphi^* \mathcal{L}_{n - d - 2}\) span the same ray in \(N^1(\overline{\mathcal{M}}_{0,n})\), so the spaces \(V_{d,n} /\!/_L \text{SL}_{d+1}\) and \(V_{n-d-2,n} /\!/_L \text{SL}_{n-d-1}\), with \(L\) the respective symmetric linearizations, have isomorphic normalizations.