9

It has been proved by Thomsen [Thomsen J. F.. “Frobenius direct images of line bundles on toric varieties” Journal of Algebra 226, no. 2 (2000)] that such a push-forward is a direct sum of line bundles. To be precise, his theorem applies to the Frobenius map on a smooth toric variety in characteristic $p$, but the same result holds for the "toric Frobenius" ...


8

No, not in general. Take $C=\mathbb{P}^1$, $L=\mathcal{O}(1)$, $p$ to be map $x\mapsto x^2$ in affine coordinates. Then $p_*L$ has rank $2$, but $$2=h^0(L)=h^0(p_*L)=h^0(\mathcal{O}(e_1))+h^0(\mathcal{O}(e_2))$$ If $e_1$ and $e_2$ were both positive, then term on the right would be at least $4$. So this is impossible. Added in response to comment. If you ...


4

You can prove this using the "Moving Lemma." Let $k$ be a field. Let $Z$ be a projective $k$-scheme that is integral of dimension $d$. Let $D\subset Z$ be a proper closed subset of $Z$. Let $C\subseteq Z$ be a closed subscheme that is integral of dimension $d-e$. Let $W_1,\dots,W_r\subseteq Z$ be integral closed subschemes of dimensions $e_1,\dots,e_r&...


4

Take $f:X\rightarrow Y$ a double covering of smooth varieties, branched along an ample divisor $D$. Then there is a line bundle $L$ such that $D$ is the zero divisor of a section of $L^2$, and $f_*\mathcal{O}_X=\mathcal{O}_X\oplus L^{-1}$, which is certainly not nef.


2

Let $W = C \times_Y X$. I imagine $f^*C$ is suitably interpreted as the chow class on $W$ given by $C \cdot X$. Write $i : C \subseteq Y$, $f': W \to C$, $i' : W \to X$. (Suppose $i$ is l.c.i. so $i^!$ makes sense, or use obstruction theories. This is automatic if $C, Y$ are smooth). Then Gysin pullback and pushforward commute, so $i^! f_* [D] = f'_* i'^![D] ...


2

The answer to the first part is no. The point is that surjectivity is not as strong a condition in this context as one might wish for. Let $X=Y=Z=[0,1]$ with the Borel $\sigma$-algebra and $\mu_X=\mu_Y=\mu_Z$ be the uniform distribution. Consider the non-surjective function $x\mapsto (x,x)$. It's push-forward is clearly not $\mu_Y\otimes\mu_Z$, it is the ...


2

OK, let me assume that your $F$ is a projective sub-bundle. Then $i_*$ is injective. Note that the blowing up setting is completely irrelevant to the situation: you just have two projective bundles $p:E\rightarrow Y$, $q:F\rightarrow Y$ and an embedding $i:F\hookrightarrow E$ over $Y$. Now the Chow group of such bundles are well-known. If $h$ is the class ...


1

Let $f:X\rightarrow Y$ be a finite separable morphism of degree two between two smooth curves. Let $R\subset X$ be the ramification divisor and $B = f_{*}R\subset Y$ the branch divisor. Then $$(det f_{*}\mathcal{O}_{X})^{2}\cong\mathcal{O}_{Y}(-B)$$ and $\mathcal{O}_{Y}(-B)$ is not nef.


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