By David Mumford

Lecture 1, what's a curve and the way explicitly do we describe them?--Lecture 2, The moduli house of curves: definition, coordinatization, and a few properties.--Lecture three, How Jacobians and theta services arise.--Lecture four, The Torelli theorem and the Schottky challenge

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**Extra resources for Curves and Their Jacobians**

**Example text**

If M E M«n + l)p, n), then M = [Mil, i = 1, ... ,p with Mi E M(n + 1, n). Let Wi(M) = span[Mi) be the row space of Mi and let Vj(M) = j Wi(M) for j = 1, ... ,po Set TO = 0, Tj(M) = dim Vj(M), ej(M) = Tj(M)i=l Tj-l(M), j = 1, ... ,p and e(M) = (e 1 (M), ... ,eP(M». We note that Tp(M) = e 1 (M) + ... + eP(M) = rankM ~ n and that 0 ~ ej(M) ~ n. 7S) ~ [ciAj)j=o, i = 1, ... ,po We label the rows of M as where Mi(A, b, C) (11· .. n + 11 12··· n + 12· .. n + Ip) so that Mi = [mji)jil and the m ji are row n-vectors.

55. 56 qS;'1]G ~ qChar(njp, 1)]. 57 G on S;'1. (Char(njp, 1),vtx ) is a geometric quotient for the action of We note that since m = 1, there is a global canonical form which we can exploit. When both m and p are greater than 1, this is impossible. So far we have shown that the theory for representations (1), (2), (3) and (4) is essentially the same as in the single input-single output case. Now suppose that f(z) = P(z)jq(z) with .. 1 1lo+b{z+ ... +b~_1zn-, ao + a1Z + ... + an_1Zn-1 pi(z) q(z) co-prime.

0,1). Again consider the set of all regular maps 'l/J: 1P'l; --* IP'% such that 'l/J(xo, xt} = ('l/Ji(XO, Xl)) with the 'l/Ji forms of degree n. We view 'l/J as a point 'I/J in IP'~On+9 (or A~On+10). 37), then 'I/J lies in a linear subspace of dimension lOn - 3. 39) and the condition that 'l/J9 be monic (or better a9n ::j:. 0), then 'I/J lies in a quasi-projective variety Va9n C IP'lOn+9. If F(z) E Rat(n, 3, 2), then 'l/Jp E Va9n and the map F --* 'l/Jp is an injective map of Rat(n, 3, 2) into Va9n • We claim it is surjective so that Rat(n, 3, 2) can be identified with the quasi-projective variety Va9n • Let 'I/J = ('l/Jo, .