An explicit algebraic representation of the Abel map, by Greg W. Anderson

This paper has appeared in International Math Research Notices 1997, No. 11, 495-521, and so the dvi version has been removed. Given a nonsingular projective curve (defined over an algebraically closed field of any characteristic) of genus $g$ and an effective divisor $G$ on that curve of positive even degree not less than $4g$, we write down a reasonably simple and explicit system of equations and inequalities the solutions of which we prove to be canonically in bijective correspondence with the divisor classes of degree equal to $\frac{1}{2}\deg G$. The paper is set at a very elementary level. Our results are formulated and proved in the context of the theory of algebraic functions of one variable. We make no use of sophisticated cohomological tools; instead we rely upon a $19^{th}$-century-style construction that we call the {\em abeliant}.

Greg W. Anderson <gwanders@math.umn.edu>