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2h

comment 
Determinant of the oriented adjacency matrix of a tree
Expanding darij grinberg's comment  it looks like the answer is $(1)^n$ where $n$ is the number of "$v$antioriented edges", i.e. the number of edges orientation of which should be switched to make all edges oriented from $v$. 
2d

awarded  Enlightened 
Nov 20 
awarded  Nice Answer 
Nov 19 
answered  Pushforward of locally free sheaves 
Nov 11 
awarded  Pundit 
Nov 10 
comment 
Analogue of BorelBottWeil for General Equivariant Vector Bundles
It follows easily from the equivalence of the category of equivariant bundles and the category of representations of Borel subgroup, since Borel is solvable. 
Nov 10 
comment 
Analogue of BorelBottWeil for General Equivariant Vector Bundles
On flag variety any equivariant vector bundle is an iterated extension of line bundles. So, its Euler characteristic can be computed by applying BBW to the factors and summing up. 
Nov 7 
answered  When does a cubic surface pass through five lines? 
Nov 6 
comment 
Vector bundles on projective varieties
But, you can always embed a vector bundle into a TWIST of a trivial bundle. 
Nov 4 
comment 
What are the higher homotopy groups of a K3 suface?
I guess you could say that $V_3$ should be the kernel of the multiplication map $S^2H^2(X,Q) \to H^4(X,Q)$, so its dimension is $b_2(b_2+1)/2  1 = 252$. Of course this is equivalent to your computation, but does not require choosing a basis and a bit simpler. 
Nov 3 
comment 
What are the higher homotopy groups of a K3 suface?
But the higher homotopy gorups of an elliptic curve is easy to find, and this is definitely a better analogy for a K3 surface. 
Oct 28 
awarded  Enlightened 
Oct 21 
comment 
Why is it so hard to compute $\pi_n(S^n)$?
Probably, to check nontriviality of the Hopf bundle one can compute its Euler class? 
Oct 14 
comment 
The linear projection of projective spaces
@abx: of course the sign was wrong, now it is corrected, thanks! 
Oct 14 
revised 
The linear projection of projective spaces
added 2 characters in body 
Oct 14 
answered  The linear projection of projective spaces 
Oct 9 
comment 
When is the Hodge diamond concentrated in $H^{n,n}$'s?
Cellular varieties have this property. But not only these. As for constructing a "related variety $X'$", I think there is no chance. What would you expect to get for a curve of positive genus? 
Oct 5 
comment 
Gequivariant coherent sheaves on BottSamelson Resolutions
Is the singularity of $X_w$ rational? If it is, then $Rf_*O = O$. 
Oct 5 
answered  Intermediate Jacobians of intersections of two quadrics 
Oct 4 
comment 
Intermediate Jacobians of intersections of two quadrics
How do you define intermediate Jacobian over an arbitrary field? As for a twist, I can say that at the level of derived category the new feature is a sheaf of Azumaya algebras appearing on the curve. So, if one can twist the Jacobian of a curve by an element of its Brauer group, then probably this is the twist you need. 