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Apr
30
comment local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$
@JérémyBlanc: Let $D \subset X$ be a class of a relative hyperplane section for $\phi$ (I think it is well defined as a Weil divisor) and take $F = \phi_*O_X(D)$. I guess $F$ is a torsion free sheaf on $P^1$, hence a vector bundle. Its rank is three and there is a birational map $X \dashrightarrow P_{P^1}(F^\vee)$. I guess it is canonical.
Apr
29
comment Smoothness of a (given) global section of a vector bundle over G(2,6)
Writing down an explicit smoothness criterion in terms of $\lambda$ might be tricky, don't expect a simple answer.
Apr
27
comment Isomorphism vs. projective equivalence: the $10$-dimensional spinor variety
@JosuaJones: Then you should use what Jason suggested.
Apr
27
comment Isomorphism vs. projective equivalence: the $10$-dimensional spinor variety
@JasonStarr: I agree, but I suspect that this is assumed and then it is much easier to explain the result.
Apr
27
comment Isomorphism vs. projective equivalence: the $10$-dimensional spinor variety
Do you assume that $O_{P^{15}}(1)\vert_X$ is the generator of $Pic(X)$?
Apr
21
comment local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$
See arxiv.org/abs/math/0509529.
Apr
19
comment A family of rank two bundles on $\mathbb CP^1$ parameterized by $H^1(O(-2n))$
No. The right answer is the following. A polynomial of degree $2n-2$ gives a symmetric bilinear form on polinomials of degree $n-1$ on the dual space. The stratification is by the rank of this bilinear form.
Apr
19
comment Reflexive sheaf and torsion free sheaf
Take the ideal of a codimension 3 subvariety.
Apr
18
comment A family of rank two bundles on $\mathbb CP^1$ parameterized by $H^1(O(-2n))$
Hint: the stratification is $GL_2$-invarriant.
Apr
14
comment Zariski open set in orthogonal grassmanian
No (over $\mathbb{C}$ positivity makes no sense). The trick is to fix one isotropic space, say $A_0$, then the components are distinguished by the parity of the dimension of intersection with $A_0$.
Apr
14
answered Zariski open set in orthogonal grassmanian
Apr
14
comment How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?
Assume you have a plane with a smooth conic in it. The conic then is isomorphic to $P^1$, and its embedding into the plane is the second Veronese. Therefore, the plane identifies with the projectivization of the symmetric square of the two-space, whose projectivization is $P^1$.
Apr
14
answered How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?
Apr
12
comment A generalization of scrolls
Another generalization is a projectivization of a vector bundle.
Apr
2
comment Polars of algebraic curves and surfaces
The relation is $F^1(x) = F(a,x,\dots,x)$. For a discussion, see "Polar covariants of plane cubics and quartics" by Dolgachev and Kanev.
Mar
17
comment Borel--Bott--Weil for the Grassmannians
Also, there is a useful extension for vector bundles corresponding to the irreducible representations of the parabolic subgroup.
Mar
12
comment Littlewood-Richardson rule for the complete flag variety: GapP complete?
Is it about complete flags in type A, or in arbitrary Dynkin type?
Mar
12
comment Littlewood-Richardson rule for the complete flag variety: GapP complete?
Don't you want to explain, what is the problem?
Mar
11
comment Hilbert scheme of quasi-projective sub-schemes
If you fix the Hilbert polynomial of the closure and of the boundary, then you can consider the flagged Hilbert scheme (parameterizing pairs $Z_b \subset Z_c \subset P^n$ of closed subschemes), and consider the family of differences $Z_c \setminus Z_b$.
Mar
9
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