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This does not answer the question, since the question concerns both genera of residual and of non reduced curves, but addresses only the issue of residual genus formulas, and in the case of $\mathbb{P …

Complex geometric view of Riemann Roch for a curve $C$: The essential Riemann Roch problem is the computation of the dimension of the vector space $H^0(D)$ where $D$ is an effective divisor on $C$. …

RRT There is a big difference in difficulty between the compact Riemann surface case and the projective curve case, for reasons already mentioned. Namely a projective curve comes equipped with a larg …

I have always felt this was due to Riemann himself: especially in: Theory of Abelian Functions, 1857. Of course the association of a Riemann surface to an algebraic curve is generally attributed to h …

Felipe's answer is much better, but what about this, just for fun: an automorphism f of X induces an automorphism of effective divisors of degree 2, i.e. of the symmetric square X^(2). This object c …

This will need expansion by a more knowledgable person, but as memory serves, it was proved by Mayer and Mumford that the closure in Ag of the locus of traditional Jacobians is the set of products of …

read walker's algebraic curves, the first few chapters, for a nice discussion of this. there you will find I believe something like e.g. that a curve of degree d with more than (1/2)(d-1)(d-2) singul …

In his paper cited above on Abelian functions, and appealing to his earlier thesis results, Riemann sketches a functor from the category of irreducible plane algebraic curves with rational maps and ra …

Expanding on Mariano's answer, my favorite examples are the ones I learned in Walker's plane algebraic curves: if the degree of C is d and there are (1/2)(d-1)(d-2) singular points, then the normaliza …

To me, excellent as the others are, engelbrekt's is the most direct answer to your question. I.e. 1) Every projective plane curve is a compact Riemann surface, essentially because of the implicit fu …

This is related to Kevin's answer but goes back to the 19th century. A Riemann surface is planar if a simple closed curve separates it. A Riemann surface of genus g becomes planar after cutting g ha …