5
votes
Accepted
Segre embedding and intersections by hyperplanes
This is a standard projective duality argument.
Let $W = H^0(\mathcal{O}_{\mathbb{P}^8}(1))$. Consider the variety $X$ of tuples
$$
(P,H_1,H_2,H_3) \in V \times W^{\oplus 3}
$$
such that $V \cap H_1 \...
5
votes
Accepted
Does the Jacobian functor respect deformations?
If I understand your question correctly, this is true. Let me be a little careful about terminology, because there are multiple notions of lift, but luckily in this case they all agree.
When talking ...
5
votes
Accepted
semiample of canonical bundle in a smooth family (Campana's proof)
Let $f:X\rightarrow \Delta $ be your family. $\ (*)$ implies that $f_*K_{X/\Delta }^{N}$ is a vector bundle on $\Delta $, with fiber $H^0(X_t, K_{X_t}^N)$ at $t\in\Delta$. The canonical homomorphism $\...
4
votes
How does Kontsevich's formality theorem apply to coherent sheaves?
Indeed, $\mathbf t$ corresponds to the $\mathcal T_{\mathrm{poly}}$ side (where the Poisson brackets live) and $\mathcal D_{\mathrm{poly}}$ corresponds to the natural generalization of associative ...
2
votes
Accepted
Lower bound for the dimension of the space of deformations $\mathrm{Defor}(f : X \to Y)$ in relative setting
This is too long for a comment.
You need some sort of hypothesis to get the existence of a versal deformation space for morphisms $f$. The most common hypothesis is that $X$ is proper over your field ...
Community wiki
2
votes
Accepted
Degeneration of curves in smooth families
I am just writing my comments as one answer. Without further hypotheses, there are counterexamples. Even without a specific example of $\mathcal{X}$, there are plenty of examples of a $K$-scheme $B_K$...
Community wiki
1
vote
Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?
Let $K$ be the fraction field of $\mathcal{O}$.
Assuming $A$ and $B$ are $\mathcal{O}$-flat, then also $A \widehat{\otimes} B$ is $\mathcal{O}$-flat (exercise), so it's enough to see that $C:=(A \...
1
vote
Non-associative deformation quantization
I figured out that in full generality this problem has no chance of leading to a different algebraic structure for which the given one is a quasi-classical limit (like it is for associative/Poisson): ...
Only top scored, non community-wiki answers of a minimum length are eligible
Related Tags
deformation-theory × 643ag.algebraic-geometry × 444
complex-geometry × 78
reference-request × 56
moduli-spaces × 44
hilbert-schemes × 41
ac.commutative-algebra × 35
algebraic-curves × 33
singularity-theory × 25
dg.differential-geometry × 24
qa.quantum-algebra × 22
flatness × 22
homological-algebra × 21
ra.rings-and-algebras × 19
galois-representations × 18
nt.number-theory × 17
rt.representation-theory × 17
arithmetic-geometry × 17
at.algebraic-topology × 16
lie-algebras × 15
mp.mathematical-physics × 15
coherent-sheaves × 15
quantization × 15
ct.category-theory × 14
algebraic-surfaces × 14