# morphisms in the construction of the moduli space of curves by mumford

Hi fellow mathematicians,

I'm just studying mumfords proof of the existence of a coarse modulispace for curves of genus $g$in his great GIT book. On page 102 (in the proof of proposition 5.2) he talks about two natural homomorphisms $h_1,h_2$. I thinking about this for quite a whole now, but I simply can't see, what exactly these morphisms do on the level of objects. Can those morphisms be written down exactly?

I would be very thankful for any hints & explanations.

Regards from Tuebingen

Wolfgang

-
Dear Wolfgang, could you maybe include the relevant definitions in your question? I'm not sure everyone interested in your question has Mumford's book at hand. – Piotr Achinger Nov 8 '12 at 17:01
On the left side those $H^0$'s are for projective space over $\mathbf{Z}$ (see item #5 on first page of Ch. 0), so tensoring with $O_S$ there amounts to the left side being the $O_S$-module of section of $O(1)$ on projective $n$-space over $S$: linear homogeneous polynomials in $n+1$ variables over $O_S$. So these maps are instances of the "base change" map $f_{\ast}(F) \rightarrow g_{\ast}(h^{\ast}(F))$ for a map $h:X \rightarrow Y$ between $S$-schemes $f:Y \rightarrow S$ and $g:X \rightarrow S$ with $F$ an $O_Y$-module (take $Y=\mathbf{P}^n_S$, $X=\Gamma_i$, $F=O_{\mathbf{P}^n_S}(1)$). – user27056 Nov 8 '12 at 23:04