16
votes
Accepted
Deformation invariance of Fano varieties
The answer is yes, in fact the following result holds.
Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$ having at most terminal, $\mathbb{Q}$-factorial ...
- 66.3k
11
votes
Accepted
Automorphism of moduli space of stable vector bundles over a curve
The moduli space of rank 2 vector bundles on $C$ with fixed determinant of odd degree is a smooth complete intersection of two quadrics in $\ \mathbb{P}^5\qquad$ (P. Newstead, Topology 7 (1968), 205-...
- 38k
10
votes
Accepted
Do all Fano threefolds have effective $c_2$?
By a theorem of Miyaoka (Theorem 6.1 in Y. Miyaoka `The Chern classes and Kodaira dimension of an algebraic variety', 1987), we have that for a Fano variety $X$, $c_2(X)\cdot H\ge 0$ for any ample ...
- 549
9
votes
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Anti-canonical divisor of a Fano variety
If you want to consider smooth (weak) Fano variety, then Fukuda has effective estimation of the birationality of anti-canonical systems for any dimension (but not optimal), see [S. FUKUDA, A note on ...
- 1,164
9
votes
Accepted
Rationality of $V_1$ fano threefold
You mean degree 6 in $\mathbb{P}(3,2,1,1,1)$. It is not rational, and its birational automorphisms are biregular. This has been proved by M. Grinenko, Mori structures on a Fano threefold of index 2 ...
8
votes
Accepted
How to compute the periodic cyclic homology of this algebra
You can use a derived version of the HKR theorem, i.e.
$HH(A) = \text{Sym}_A^\bullet(\mathbb{L}_A[1])$ where $\mathbb{L}_A$ is the cotangent complex (over $k$). I'm not sure about a reference though.
...
- 1,423
7
votes
Accepted
Torsion in the cohomology of Fano varieties of lines
Here is an expanded version of my comments. Let's work over the complex numbers which I suppose is assumed in the question. Let $K_0(Var)$ be the Grothendieck ring of varieties (see e.g. https://arxiv....
- 2,300
7
votes
Accepted
A Fourier-Mukai equivalence between non trivial component of cubic threefold and degree 14 prime Fano threefold
This is a rational quartic curve.
Indeed, the FM kernel is induced by the HPD kernel, which is, essentially, the locus $\mathbf{Z}$ of pairs $(U,y)$, where $U$ is a 2-dimensional subspace in the fixed ...
- 39.3k
6
votes
Accepted
Formula for genus of a Fano variety
It's not clear from your question if you really want a reference (as per the first line) or the formula itself (as per what is written after "Question"). But the computation of the genus is ...
- 887
6
votes
Accepted
Dual of stable vector bundle on a Fano threefold
For any vector bundle $E$ of rank 2 there is an isomorphism
$$
E^* \cong E \otimes \det(E^*).
$$
If $\det(E) = \mathcal{O}(1)$, this boils down to $E^* \cong E(-1)$.
- 39.3k
6
votes
Accepted
Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold
$\mathcal{N}_{X|G}$ is in your case $F|_X$, where $F=O_G(1)^2 \oplus O_G(2))$. In order to compute the first two spaces, you can simply use the Koszul complex for X $$ 0 \to det(F^{\vee}) \to \wedge^2 ...
- 776
6
votes
Accepted
Semi-orthogonal decomposition for maximally non-factorial Fano threefolds
If you assume that $\mathcal{P}$ and $\mathcal{P'}$ are semiorthogonal, this is true. The easiest way to see this is by looking at the singularity category. If $X$ has one node, (the idempotent ...
- 39.3k
6
votes
How to compute the periodic cyclic homology of this algebra
There's a spectral sequence starting with, for example, Hochschild cohomology of the cohomology of a DGA algebra $A$ and converging to the Hochschild cohomology of $A$, $E_2^{p,q}=H\!H^*H^*(A)\...
- 16.2k
5
votes
Accepted
Minimal embeddings of certain Fano varieties with Picard number one
This is definitely not true; just consider two smooth cubic threefolds in $\mathbb{P}^4$.
Of course many other examples exist, e.g. Fano hypersurfaces in projective space which are not quadrics nor ...
- 21.1k
5
votes
Mirror symmetry for blowups of the projective plane
It depends on the positions of the points that you blow up. If you blow up respectively $p,q$ and $r$ points on the 3 irreducible components of the toric divisor $D\subset\mathbb{CP}^2$, with $p,q,r\...
- 3,187
4
votes
Accepted
Fano blow ups of $\mathbb CP^n$
Note. My comment above was wrong; I had the wrong denominators. When you correct the denominators, the formula gives an asymptotic result.
In my comment I wrote the wrong formula for the denominator ...
4
votes
Accepted
Can free rational curves lift to ramified covers of Fano varieties?
After some helpful conversations with Johan de Jong, I came up with an example. After finding it I realize I probably should have figured it out earlier.
In fact, $\operatorname{Hilb}^2(\mathbb P^n)$ ...
- 148k
4
votes
Accepted
The locus of lines intersecting with another fixed line on a Fano threefold
Question 1. Let $I(Y) \subset \Sigma(Y) \times \Sigma(Y) \cong \mathbb{P}^2 \times \mathbb{P}^2$ be the incidence scheme (parameterizing pairs of intersecting lines). Then $I(Y) \cong \mathrm{Fl}(1,2;...
- 39.3k
4
votes
Accepted
Do non-compact Fano manifolds exist?
By the Bonnet Myers theorem, bounded positive Ricci curvature and complete Riemannian metric forces compact. David Wraith once explained to me that if the Ricci decays more slowly than quadratically ...
- 26.3k
4
votes
Accepted
liftability of isomorphism of curves in $P^3$
An isomorphism $f \colon C \to C'$ of (linearly normal) projective curves lifts to an isomorphism of their ambient spaces if and only if
$$
f^*L' \cong L,\tag{*}
$$
where $L$ and $L'$ are the line ...
- 39.3k
3
votes
Accepted
Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$
The polytopes you are interested in are related by sequences of combinatorial mutations, as described here and here. If two polytopes $P_1$ and $P_2$ are related by a combinatorial mutation, then ...
- 1,306
3
votes
Minimal embeddings of certain Fano varieties with Picard number one
Even simpler example: Grassmannian Gr(3,6) and $X=Gr(2,7) \cap H$, an hyperplane section of Grassmannian Gr(2,7).
They are prime (by Lefschetz), have the same dimension (9), lies in the same space $\...
- 776
3
votes
Quotient of a Fano variety by a torus
When the Fano variety is smooth, the field is $\mathbb{C}$ and the torus has complex dimension 1 (along with some the mild hypothesis that the (restriction of the action to $S^{1} \subset \mathbb{C}^{*...
- 6,995
3
votes
Accepted
How to check that exceptional sequence of vector bundles on Fano variety is helix foundation
It is not enough to check only $E_0$, but checking $E_0,\dots,E_n$ is enough.
There are many ways of proving fullness, most of them require first to construct some more objects from the considered ...
- 39.3k
3
votes
Fiberwise compactification of a LG model
For the mirror of $\mathbb{CP}^2$, a smooth fiber of $W$ is a 3-punctured torus, and you can compactify the fibers by filling in 1, 2 or 3 punctures. This corresponds, in $\mathbb{CP}^2$, to different ...
- 3,187
3
votes
Analogy of a Fano manifold with anticanonical divisor
I'd say it's closer to an oriented manifold with corners (corners happening where the divisor is singular), or even that times a coefficient. In these papers Khesin, Rosly, and later Thomas build a ...
- 27.8k
2
votes
Pseudo-automorphisms on Fano varieties
It seems to me that the Fano variety $X$ does not need to be smooth. It is enough to have that $-K_X$ is $\mathbb{Q}$-Cartier. Indeed, in this case we can embed $X$ in a projective space $\mathbb{P}^n$...
- 8,998
2
votes
Accepted
Negative Definite Fano Manifolds
Such manifold are called of general type (at least in the projective case). If you look at compact Riemann surfaces, then the only Fano variety is $\mathbb{P}^1$ while curves of general type have ...
- 196
2
votes
Accepted
A short exact sequence on del Pezzo threefold and Gushel-Mukai
These sequences do not exist, because the kernel of an epimorphism of locally free sheaves is itself locally free, while the ideal sheaf of a curve on a threefold is not locally free.
Instead, there ...
- 39.3k
2
votes
$K3$ surfaces in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$
In the generic case you'll have $\text{NS}(S)\otimes\mathbb R\cong\mathbb R^3$ and you can compute the $3$-by-$3$ matrix for the action of the three involutions $i_1,i_2,i_3$ on, say, a basis ...
- 47.4k
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