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Algebraic Geometry

By David Mumford

From the studies: "Although numerous textbooks on smooth algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the purple booklet of types and schemes, now as prior to probably the most first-class and profound primers of recent algebraic geometry. either books are only actual classics!" Zentralblatt

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Consequently the map . . . sending an endomorphism q0 of X to its dual ~ defines an anti-involution on End(X). In fact . . . is just the Rosati involution of (X, L). Let Y be an abelian subvariety of X of dimension g' and t: Y ~-+ X the canonical embedding. The line bundle t*L defines a polarization on Y and the corresponding isogeny ¢y: Y -+ ~', y ~ t*yL*L @ L*L-1, fits into the commutative diagram y Cr t (1) Since ¢ y is an isogeny, it has an inverse ¢y1 in HomQ(Y, Y) = Horn(Y, Y) ® Q. Let e(Y) denote the smallest positive integer, such that ~bv := e(Y)¢yl: Y --+ y is a homomorphism.

Denote by (3:, Z) a pair of complementary abelian subvarietie~ with d i m Y >__d i m Z . If LIZ is of type ( d l , . . , d r ) , then LIY is of type (1 . . . 1 , d l , . . , d r ) . 27 4. N o r m - e n d o m o r p h i s m s associated to a covering of curves Let f: C ~ C' be a morphism of degree n of smooth projective curves C of genus g and C' of genus g'. Denote by J = J ( C ) and J ' = J ' ( C ' ) the corresponding Jacobians with canonical principal polarizations defined by line bundles L and L'.

Aknowledgements. This research was started during a visit of C. Ciliberto and M. Teixidor at the Department of Mathematics of the Brown University in 1987 and concluded during a visit of C. Ciliherto at the Departments of Mathematics of the Universities of Brandeis and Harvard in 1989. C. Ciliberto and M. Teixidor would like to express their gratitude to all, institutions and colligues, w h o m a d e it possible for them to enjoy these visits. Furthermore the three authors are grateful to P. Pirola w h o pointed out a mistake in an earlier version of this paper.

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