Different ways to construct maps and the tensor products of line bundles

2011-02-04T14:35:42Z

Let $C$ be a curve. Then I know of two ways to create morphisms. To get morphisms from $C$, take a line bundle of any degree $L$ and use the linear system it determines to get a map into projective space, which may or may not be injective, so you get a map to another curve. To get a morphism to $C$, a particular case is to take a line bundle $\mu$ of order 2, which gives us a curve $\tilde{C}$ as a double cover of $C$.

How do these two methods interact? For instance, given a curve $C$, and the tower $\tilde{C}\to C\to \bar{C}$ with the first map given as a double cover by $\mu$, and the second map given by a line bundle $L$, presumably the geometry of this tower is connected to the line bundle $L\otimes\mu$, but it's not obvious to me how to connect the two notions to get any actual information.

Answer by Francesco Polizzi for Different ways to construct maps and the tensor products of line bundles

2011-02-04T20:19:17Z

A possible answer is the following. Assume

$h^0(L)=n, \quad h^0(L \otimes \mu)=m$,

set

$\phi \colon \tilde{C} \to C, \quad \psi \colon C \to \bar{C} \subset \mathbb{P}^{n-1}$

and let $f=\psi \circ \phi \colon \tilde{C} \to \mathbb{P}^{n-1}$ be the composition.

Then $f^*\mathcal{O}_{P^n}(1)=\phi^*L$. On the other hand, by projection formula we have

$h^0(\phi^*L)=h^0(L) + h^0(L \otimes \mu)=n+m$,

and this shows that the complete linear system $|\phi^*L|$ induces a map $g \colon \tilde{C} \to \mathbb{P}^{n+m-1}$. Moreover, the map $f$ is obtained by composing $g$ with the projection from the linear subspace $\mathbb{P}^{m-1} \subset \mathbb{P}^{n+m-1}$ corresponding to the natural inclusion $\phi^* H^0(L \otimes \mu) \subset H^0(\phi^*L)$ .

The easiest example is perhaps the following. Assume that $C$ is a genus $2$ curve and take $L=K_C$. Then $\bar{C}=\mathbb{P}^1$, $\tilde{C}$ is a hyperelliptic (this can be proven) genus $3$ curve and $\phi^*L=K_{\tilde{C}}$ . We have

$h^0(L)=2, \quad h^0(L \otimes \mu)=1$

and the map $g \colon \tilde{C} \to \mathbb{P}^2$ is precisely the canonical map of $\tilde{C}$, which is a double cover of a conic $D \subset \mathbb{P}^2$. The map $f \colon \tilde{C} \to \mathbb{P}^1$ is a quadruple cover, obtained by composing $g$ with the projection of $D$ from the point in $\mathbb{P}^2$ corresponding to the non-zero section of $L \otimes \mu$.

Different ways to construct maps and the tensor products of line bundles - MathOverflow

Different ways to construct maps and the tensor products of line bundles

Answer by Francesco Polizzi for Different ways to construct maps and the tensor products of line bundles