[r] (J 0 (C))}, and UG [r] [r] (J 0 (C))}.The group UG • [r] is of index two in UG [r] and we have a natural homomorphism:where η ∈ J 0 (C) is an arbitrary line bundle with the property η ⊗r = det E ⊗ L −1 0 . If in addition r | 2d, then the homomorphism (0.3) can be extended toThe commutative diagram (0.2) suggests that the subgroups (0.3) (or (0.4) in the case r | 2d)will not differ too much from the full automorphism group, if the moduli space SU (r, L 0 ) doesn't have excess automorphisms. The variety SU (r, L 0 ) has a lot of advantages. It is a Fano variety of Picard number one and by a theorem of Narasimhan and Ramanan, [N-R 2], H 0 (SU (r, L 0 ), T SU (r, L 0 )) = 0 unless g = 2, r = 2 and d even. Therefore, in general, Aut(SU (r, L 0 )) is finite and contains the group J 0 (C) [r]. In the case r = 2, odd degree and g = 2 the group Aut(SU s (2, L 0 )) was described by Newstead [Ne] as a consequence of his proof of the Torelli theorem for the variety SU s (2, L 0 ).To generalize his result we adopt a different viewpoint -we use Hitchin's abelianization to prove our main theoremTheorem A Let C be a curve without automorphisms. Then the automorphism group of the moduli space SU (r, L 0 ) can be described as follows 1. If r ∤ 2d, then the natural mapis an isomorphism, and 2. If r | 2d, then the natural map J 0 (C)[r] ⋊ Z/2Z −→ Aut(SU (r, L 0 )) (µ, ε) −→ (E → δ ε (E) ⊗ µ)is isomorphism for r ≥ 3 and has kernel Z/2Z for r = 2.
Abstract. We determine the cycle classes of effective divisors in the compactified Hurwitz spaces H d,g of curves of genus g with a linear system of degree d, that extend the Maroni divisors on H d,g . Our approach uses Chern classes associated to a global-to-local evaluation map of a vector bundle over a generic P 1 -bundle over the Hurwitz space.
Abstract.For a curve with general moduli, the Neron-Severi group of its symmetric products is generated by the classes of two divisors x and 6 . In this paper we give bounds for the cones of eifective and ample divisors in the x8-plane.
Abstract. By using the Grothendieck-Riemann-Roch theorem we derive cycle relations modulo algebraic equivalence in the Jacobian of a curve. The relations generalize the relations found by Colombo and van Geemen and are analogous to but simpler than the relations recently found by Herbaut. In an appendix due to Zagier it is shown that these sets of relations are equivalent.
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