MR0435074 (55 #8036) Reviewed
Griffiths, Phillip A.
Variations on a theorem of Abel.
Invent. Math. 35 (1976), 321–390.
14C25 (14C30 32J25)More links
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This is the first in a series of papers and is of a partly expository nature. The presentation is didactically chosen so as to assume practically nothing in the beginning and to prove as much as possible from few (admittedly deep) theorems. Much of the material covered is not new, only the viewpoint is different. The paper is divided into three parts, each of which will be considered separately.
   Part I is devoted to the origins of Abel's theorem. Nowadays Abel's theorem would be stated as follows: Suppose that C is a smooth algebraic curve; then the divisor Γ=P1+P2+⋯+Pk on C is linearly equivalent to the divisor Γ′=Q1+⋯+Qk if and only if ∑ν∫PνP0ω=∑ν∫QνP0ω (modulo periods) for all regular (i.e., holomorphic) 1-forms on C. (Here P0 denotes some fixed reference point on C.)
   The author, however, takes as his starting point the following formulation: Let C be an algebraic plane curve (not necessarily smooth) and Dt a family of plane algebraic curves meeting C in variable points Pν(t)(ν=1,⋯,k); suppose ω is a rational 1-form on P2 not identical to ∞ on C; then the Abel sum u(t)=∑ν∫PνP0ω is of the form: u(t)=R(t)+∑νlogSν(t), where R and S are rational in t.