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.  

 

 

 

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