# line bundles, finite morphisms between curves

These posts are for my benefit and to save me time I’m going to start writing $O$ instead of $\mathcal{O}$.

Let $X$ be an integral scheme.  The long exact sequence associated to

$0 \to O_X^* \to K^* \to K^*/O_X^* \to 0$

gives that $\Gamma(K^*/O_X^*) \to H^1(O_X^*) \to H^1(K^*)$ is exact, on the other hand, the last group is the higher cohomology of a flasque sheaf on an irreducible space, hence the group is zero, and the first map is a surjection.  Recalling the that $\mbox{Pic}(X) \cong H^1(O_X^*)$ I conclude any line bundle is of the form $O_X(D)$ for some Cartier divisor $D$.

Recall a sheaf is a generated by global sections if the images of the global sections generate the stalk at every point.  For a line bundle, this says for every point p there is a global section that doesn’t vanish at p.  Also recall that $\Gamma(O(D-p)) \subset \Gamma(O(D))$ consist of the global sections that vanish at p.  Putting this all together, I get $L$ is generated by global sections iff for all p, $h^0(L(-p)) < h^0(L)$.  Here’s a related thm:

Thm: If $X$ is a curve of genus $g \ge 1$ then the canonical sheaf $\omega$ is generated by global sections.

proof: Say there exists a point p such that $h^0(\omega(-p)) = h^0(\omega) = h^1(O_X) = g$.  Then applying R-R to the line bundle $O(p)$, I get

$h^0(O(p)) = h^0(\omega(-p)) + 1 + 1 - g = 2$

This implies $g(X) = 0$ which is a contradiction. QED

Next little bit of lecture concerned the following setup.  $f \colon X \to Y$ is a finite morphism between smooth curves ( I think it might be enough for X to be smooth as long as Y is at least integral). If $y \in Y$ is a closed point, then

$f^{-1}(y) = \sum_{x \mapsto y} e_x \cdot x$

where the $e_x$ are determined via

$f^{-1}m_yO_X = \prod_{x \mapsto y} m_x^{e_x}$

Recall $m_x$ is principle.  Define $f_x = [k(x):k(y)]$.

Prop: With the above setup, $sum_x e_xf_x = deg(f)$.

proof: The degree of a finite morphisms is defined to be $[K(X):K(Y)]$.  Now using a result in Hartshorne like $\deg f^{-1}(y) = \deg f^*(y) = deg(f) \cdot \deg y$ or something similar, plus using cohomology with base change I can conclude $h^0(X_y,( O_X)_y)$ is constant so in fact $f_*O_X$ is locally free on Y, and so

$\deg(f) = rk f_*O_X = \dim_{k(y)} (f_*O_X)_y \otimes_{O_{Y,y}} k(y)$

where $y$ is some closed point.  Continuing to compute,

$= \dim_{k(y)} (f_*O_X)_y/m_y(f_*O_X)_y = \dim_{k(y)}f_*(O_X/f^{-1}m_yO_X)_y$

The commutativity of f lower star and quotient can be checked for every open set (i.e. they give the same group for every open set, so they are the same).  Continue computing:

$= \dim_{k(y)} \varinjlim_U O_X(U)/\prod_x m_x^{e_x} O_X(U)$

where $U$ ranges over a subset of the open sets in $X$ that contain all the points mapping to y.  Using the chinese remainder theorem, this can be expressed as

$= \sum_{x \mapsto y} e_x \dim_{k(y)} k(x) = \sum_x e_x f_x$.

QED.

Here something else using the set up as above.  Let $D$ be a Cartier divisor on Y.  Consider the associated closed subscheme of codimesion 1 and define $f^*(D) := f^{-1}(D)$ considered as a closed subscheme in $X$.  Then $f^*(D)$ as the fibered product of $f \colon X \to Y$ and the inclusion $j\colon D \to Y$.  Let $i \colon f^*(D) \to X$ be the inclusion map.  Using flat base change it follows that $f^*j_*O_D \cong i_*f^*O_D \cong i_*O_{f^*(D)}$.  Now, $f^*O_Y \cong O_X$.  Putting this all together (and using that tensoring with $f_*O_X$ is exact) and starting with the s.e.s $I_D \to O_Y \to O_D$ I get that

$0 \to f^*I_D \to O_X \to i_*O_{f^*D} \to 0$

is exact.  That is, $f^*I_D$ is the ideal sheaf of $f^*(D)$.  Here’s another construction: $f_* \colon Div(X) \to Div(Y)$.  On points it is $f_*(x) = f_x \cdot f(x)$.  It can be check that $f_*f^*(y) = deg(f) \cdot y$.

Here’s another theorem.

Thm: For a curve $X$, $\omega$ is not very ample iff there exists a line bundle of degree 2 with $h^0(L) = 2$ iff there exists a degree 2 map to $\mathbb{P}^1$.

proof: Using the criterion for a line bundle to define a closed immersion, I conclude that $\omega$ is not very ample iff there exists $D$ of degree 2 s.t. $h^0(\omega(-D) = g-1$ so R-R says $h^0(D) = 2$.  But this implies there is a nonconstant function with poles only at two points (assuming g(X) is not zero) which gives a degree 2 map, and this argument can be reversed.  QED.

Ogus also talked about wild and tame ramifications and hyperelliptic curves and Hurwitz thm and Clifford’s theorem, but I didn’t have notes on this stuff, but it does appear in Hartshorne.