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\(sl_{n}\) level 1 conformal blocks divisors on \(\overline{M_{0,n}}\). (English) Zbl 1271.14034

Let \(\overline{\mathcal M}_{g,n}\) be the Deligne-Mumford moduli space of stable curves of genus \(g\) with \(n\) marked points. Consider a a simple complex, finite-dimensional, Lie algebra \(\mathfrak g\), a positive integer \(\ell\) (the level) and an \(n\)-tuple of dominant integral weights \((\lambda_1, \dots, \lambda_n)\) with level of \(\lambda_i \leq \ell\) for all \(i\). The Wess-Zumino-Witten model of conformal field theory can be thought of as defining vector bundles on \(\overline{\mathcal M}_{g,n}\) whose fibers are the so-called vector spaces of conformal blocks, with ranks computed by the Verlinde formula. In this paper the authors consider \(\overline{\mathcal M}_{0,n}\), with \(\mathfrak g = \mathfrak{sl}_n\), \(\ell = 1\) and \(n\)-tuples \(\omega = (\omega_{j_1}, \dots, \omega_{j_n})\), where \(\omega_1, \dots, \omega_{n-1}\) are the fundamental weights. The divisor associated to the corresponding conformal block bundle is denoted \(D^n_{1, \omega}\), or simply \(D^n_{1, j}\) when \(\omega = (\omega_{j}, \dots, \omega_{j})\) is \(\mathbb S_n\)-invariant. The authors study these divisors. It is shown that the divisors \(D^n_{1, j}\) for various \(j =2, \dots, [\frac n2]\) span extremal rays of the cone of symmetric nef divisors. Evidence pointing to the validity of this result was first obtained in the following way: recurrence formulae for the Chern classes of conformal block divisors were given in [N. Fakhruddin, Contemp. Math. 564, 145–176 (2012; Zbl 1244.14007)]. These formulae were implemented in a package of Macaulay and subsequent experimentation suggested the above mentioned fact. The authors also show that the morphisms from \(\overline{\mathcal M}_{0,n}\) to projective varieties defined by the divisors \(D^n_{1, \omega}\) factor through natural birational contractions defined in [B. Hassett, Adv. Math. 173, No. 2, 316–352 (2003; Zbl 1072.14014)].

MSC:

14H10 Families, moduli of curves (algebraic)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14L24 Geometric invariant theory
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