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Let $k$ be an algebraically closed field. Any finite $k$-morphism $P^1_k\rightarrow P^1_k$ is flat (miracle flatness) and surjective on the underlying spaces. Therefore, the pushforward of a coherent locally free sheaf is coherent locally free (on $P^1_k$, such sheaves can be described by a finite sequence of integers using the fact that Picard rank is 1 and there is Birkhoff--Grothendieck splitting).

Assume we have a finite $k$-morphism $P^1_k\rightarrow P^1_k$ such that the inverse image of the generic point has cardinality $n\geq 2$. Which sheaves can we get as the pushforward of a locally free sheaf of rank 1?

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  • $\begingroup$ I believe that a finite morphism sends closed points to closed points. Any point on a curve is either closed or generic. The question does not make much sense right now. Maybe you mean that the maximum cardinality of a fiber of the morphism is $n\geq 2$? $\endgroup$
    – user138661
    Apr 24 '19 at 3:33
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    $\begingroup$ Welcome new contributor! Flatness is not a miracle in this case, it is simply the classification of f.g. modules over a PID. $\endgroup$ Apr 24 '19 at 4:03
  • $\begingroup$ There are non-flat maps from $P^1$ to $P^1$, namely compositions $P^1 \to Spec(k) \to P^1$. $\endgroup$
    – Sasha
    Apr 24 '19 at 4:23
  • $\begingroup$ @Sasha they are not quasi-finite though (if you are over an algebraically closed field). $\endgroup$
    – user138661
    Apr 24 '19 at 4:31
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Write $$ f_* \mathcal{O}(m) = \bigoplus_{k\in\mathbb{Z}} \mathcal{O}(k)^{\alpha(m, k)}. $$ We want to compute the multiplicities $\alpha(m,k)$. We have $f^* \mathcal{O}(k) = \mathcal{O}(nk)$ where $n = \deg f$, so the projection formula gives $$ (f_* \mathcal{O}(m)) \otimes \mathcal{O}(-k) = f_* \mathcal{O}(m-nk),$$ and hence $\alpha(m,k) = \alpha(m-nk, 0)$. If $S(x) = \sum_m (m+1)x^m = 1/(1-x)^2$ and $A(x) = \alpha(m,0) x^m$, then applying $h^0(-)$ to both sides of the first displayed formula, multiplying by $x^m$ and summing over $m\in\mathbb{Z}$ gives $$ S(x) = A(x)\cdot S(x^n).$$ Thus $$ \alpha(m,k) = \left(\text{coefficient of }x^{m-nk}\text{ in } S(x)/S(x^n) = (1+x+\cdots + x^{n-1})^2\right). $$ For example, $$f_* \mathcal{O} = \mathcal{O} \oplus \mathcal{O}(-1)^{n-1}.$$

(The above argument appears in my paper "Frobenius Push-Forwards on Quadrics", and works similarly for $\mathbb{P}^N$. The first place I know where these pushforwards are computed is the paper "Frobenius direct images of line bundles on toric varieties" by J. F. Thomsen)

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  • $\begingroup$ I will never get sick of this calculation. Follow up (please don't feel obligated to reply!): is there a perspective (or even a calculation for a particular $f$) from which $f_* \mathcal{O}_X = \mathcal{O}_X \oplus \mathcal{O}(-1)^{n-1}$ is "obvious"? $\endgroup$
    – cgodfrey
    Apr 24 '19 at 5:07
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    $\begingroup$ @cgodfrey Yes: if $f$ is the map $(x^n\colon y^n)$ (and say if $n$ is prime to the characteristic, otherwise you need to be more careful), then $f$ is a cyclic covering with group $\mu_n$ ramified along $D = 0+\infty$. The usual theory of cyclic coverings gives $f_* \mathcal{O} = \bigoplus_{i=0}^{n-1} \mathcal{O}(\lfloor -\frac{iD}{n} \rfloor)$. Now the $i=0$ summand is $\mathcal{O}$, and the other summands are $\mathcal{O}(-1)$. $\endgroup$ Apr 24 '19 at 19:07
  • $\begingroup$ Ah perfect, thank you very much! $\endgroup$
    – cgodfrey
    Apr 24 '19 at 22:00
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Here is another way to view this.

Let's look at local sections of $\mathbb{P}^1_w$. Cover $\mathbb{P}^1_w$ by affine charts $U_0$ and $U_1$. Then $\mathcal{O}_{\mathbb{P}^1_w}$ and $\mathcal{O}_{\mathbb{P}^1_w}(-1)$ can be viewed as following pictures:

$$ \rlap{\underbrace{\phantom{\cdots w^{-3}, w^{-2}, w^{-1}}}_{U_0}} w^{-3}, w^{-2}, w^{-1}, \overbrace{1, w , w^2, w^3, \cdots}^{U_1} $$ $$ \underbrace{\cdots w^{-3}, w^{-2}, w^{-1}, 1\,}_{U_0} , \, \overbrace{ w , w^2, w^3, \cdots}^{U_1} $$

For example, the local sections of $\mathcal{O}_{\mathbb{P}^1_w}$ over Spec$k[w]$ are $1, w, w^2,\cdots$ and over Spec$k[w^{-1}]$ are $1, w^{-1}, w^{-2},\cdots$. So the global section is only $k$.

Now suppose we have the morphism given by $z=w^3$. Then we can decompose $\mathcal{O}_{\mathbb{P}^1_w}$ as following three:

$$ \rlap{\underbrace{\phantom{\cdots w^{-9}, w^{-6}, w^{-3}}}_{U_0}} w^{-9}, w^{-6}, w^{-3}, \overbrace{1, w^3 , w^6, w^9, \cdots}^{U_1} $$ $$ \underbrace{\cdots w^{-8}, w^{-5}, w^{-2}}_{U_0}, \, \overbrace{ w^{1}, w^4 , w^7, \cdots}^{U_1} $$ $$ \underbrace{\cdots w^{-7}, w^{-4}, w^{-1}}_{U_0}, \, \overbrace{ w^{2}, w^5 , w^8, \cdots}^{U_1} $$ Hence $$f_* \mathcal{O}_{\mathbb{P}^1_w} = \mathcal{O}_{\mathbb{P}^1_z} \oplus \mathcal{O}_{\mathbb{P}^1_z}(-1)^{2}.$$

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  • $\begingroup$ I didn't understand completely what the braces signify, but I think that this is quite similar to the argument on p. 2 of J. F. Thomsen "Frobenius Direct Images of Line Bundles on Toric Varieties" (J. Algebra 226 (2000)). $\endgroup$ Apr 22 at 5:26
  • $\begingroup$ I'm going to echo Piotr's comment and add that the first two displays are identical, which doesn't help. $\endgroup$
    – user347489
    Apr 22 at 6:15
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    $\begingroup$ Okay, I just learned how to use phantom to get overlapped brackets in Latex. $\endgroup$
    – WWK
    Apr 22 at 9:50

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