Monday, 25 July 2011

ag.algebraic geometry - theta divisor on a principally polarized abelian variety

This is true. For A/mathbbC an abelian variety, L an ample line bundle on A, then any line bundle
MinoperatornamePic0(A) -- over mathbbC, this is equivalent to having first Chern class zero -- is of the form TxLotimesL1 for some xinA. (e.g. Theorem 1 on p. 77 of Mumford's Abelian Varieties).



Applying this theorem with L=L(D2), M=L(D1)L(D2), we get that



L1L2=Tx(L2)L2, so



L1=Tx(L2).



So D1 and x+D2 (meaning translation of D2 by x!) must be linearly equivalent, but by your assumption h0(L(D1))=h0(L(D2))=1, they are each the unique effective
divisors in their linear equivalence classes, so we must have D1=x+D2.

No comments:

Post a Comment