8
votes
Accepted
Pushforward of a very ample line bundle on a curve to $\mathbb{P}^1$
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{...
- 35.2k
8
votes
Accepted
Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?
It essentially is "never" fully faithful. In your question, you implicitly assume that $i_*$ preserves perfect complexes - this is a restriction, but not as severe as fully faithfulness (e.g....
- 15.8k
5
votes
Accepted
Surjectivity of pushforward on image
It is true; the trick is to use the von Neumann-Jankov measurable selection theorem to construct a right inverse of $\Phi$ on $\Phi(\mathcal X)$.
The result is essentially Lemma 2.2. of [Varadarajan, ...
5
votes
Behavior of divisors under push forward and pull back
Of course, just choose a point on each (divisorial) component of the exceptional locus of $f$, then for each of these points the divisors in $|H|$ passing through form a hyperplane in $|H|$, and any ...
- 39.3k
4
votes
Accepted
Pushforward of curves
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\...
3
votes
Accepted
Change of coordinates for coends
$\require{AMScd}$First of all, I suspect that "$p$ has set fibers and it is surjective" means that it is a discrete fibration. In that case, $p$ corresponds to a functor $G_p : C \to Set$ (...
- 13.6k
2
votes
Derived pushforward of a projection
I think the idea is that you have to keep in mind what happens with manifolds, if you have something like an oriented vector bundle of dimension n $\pi:E\to M$ then there is a pushforward of ...
- 1,482
2
votes
Accepted
Pushforward of measure under surjective map
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=\...
2
votes
Accepted
Gysin map for projective sub-bundles of exceptional divisors
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 $...
- 38k
1
vote
A question about pushforward measures and continuous Borel isomorphisms
This is a very good (and also well studied) question, especially for homeomorphisms of measures.
For example, the Haar measures on the zero-dimensional compact groups $\mathbb Z_2^\omega$ and $\mathbb ...
- 41.8k
1
vote
Accepted
Rewriting PDE as "push-forward"
Let's focus on $N=1$ only, the case $N>1$ is just a tensorization of the argument below.
The whole argument is actually unrelated to the specific (aggregation-diffusion) PDE or gradient flow: As ...
- 5,380
1
vote
Push-forward of divisors and intersections
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 $i^!$...
- 1,084
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