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 \...
Sasha's user avatar
  • 37k
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 ...
R. van Dobben de Bruyn's user avatar
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 $\...
abx's user avatar
  • 37.1k
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 ...
Severin Barmeier's user avatar
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 ...
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 $B_K$...
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 \...
Satan's Minion's user avatar
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): ...
Vladimir Dotsenko's user avatar

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