Hecke curves and hitchin discriminant

Let C be a smooth projective curve of genus g greater than or equal to 4 over the complex numbers and SUCs (r, d) be the moduli space of stable vector bundles of rank r with a fixed determinant of degree d. In the projectivized cotangent space at a general point E of SUCs (r, d), there exists a distinguished hypersurface SE consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety C-E consisting of vectors tangent to Hecke curves in SUCs(r,d) through E. Our main result establishes that the hypersurface SE and the variety CE are dual to each other. As an application of this duality relation, we prove that any surjective morphism SUCs (r, d) --> SUCs, (r, d), where C-1 is another curve of genus g, is biregular. This confirms, for SUCs(r, d), the general expectation that a Fano variety of Picard number 1, excepting the projective space, has no non-trivial self-morphism and that morphisms between Fano varieties of Picard number 1 are rare. The duality relation also gives simple proofs of the non-abelian Torelli theorem and the result of Kouvidakis-Pantev on the automorphisms of SUCs(r, d). (C) 2004 Published by Elsevier SAS