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2d
comment Locally free sheaves and flat families of projective scheme
even with this --- take $X = Y = {\mathbb A}^1$. Then the only nonclosed point is the generic point and for it the tensor product is zero.
2d
comment Locally free sheaves and flat families of projective scheme
No. Take $f$ to be identity and $F$ to be the structure sheaf of a point.
2d
comment Does a fully faithful functor between triangulated categories induce embedding of their Grothendieck groups?
Density is a very strong assumption!
2d
comment Deformation of curves and closed immersions
My impression was that the question asked whether the embedding of a special curve can factor through a LINEARLY embedded ${\mathbb P}^3$, and in this example it is embedded via the second Veronese embedding.
2d
comment Vector bundle is semistable if only if it's pull back is semistable?
Simplest counterexample is $X = {\mathbb P}^2$, $D$ a line, $E = T_X$.
Apr
10
comment Are schemes which agree on open set and its complement equal? - w/ applications to initial ideals/tropical basis
An Artin subscheme of length 2 is a complete intersection of two hypersurfaces (say $x = y^2 = 0$), so I guess this still gives you a counterexample.
Apr
10
answered Are schemes which agree on open set and its complement equal? - w/ applications to initial ideals/tropical basis
Apr
4
comment What is the applications of the dg-enhancements of derived categories of sheaves
Whenever you want to perform some nontrivial construction with triangulated categories (e.g. gluing) you have to use enhancements.
Apr
2
comment Connected vs Irreducible Subvarieties
Sometimes smoothness is easier to check than normality, and of course it also implies irreducibility of connected components.
Mar
28
comment Coherent sheaves on $\mathbb C^2$ and commuting matrices
An upper triangular matrix is nilpotent if and only if its diagonal is zero. A subquotient is $V_k/V_l$.
Mar
28
comment Coherent sheaves on $\mathbb C^2$ and commuting matrices
Choose a basis in which both $A$ and $B$ are uppertriangular. Let $a_1,\dots,a_n$ be the diagonal entries of $A$ and $b_1,\dots,b_n$ those of $B$. Then the support is $(a_1,b_1) + \dots + (a_n,b_n)$. A subquotient means that on $\mathbb{C}^n$ there is a filtration preserved by $A$ and $B$ such that $(A',B')$ is its subquotient.
Mar
28
answered Coherent sheaves on $\mathbb C^2$ and commuting matrices
Mar
28
comment Root in positive Weyl chamber
@shu: This is a standard argument which does not depend on the classification. I would advise you to read any book on root systems.
Mar
28
comment Root in positive Weyl chamber
I guess there are only two cases when there is a root in $K$. One is $A_2$ with the highest root being $\rho$. The other is $A_1$ with the highest root being $2\rho$.
Mar
27
comment Root in positive Weyl chamber
It is easy to classify $R \cap \overline{K}$. In the simply-laced case this set consists just of one root (the highest root). In the non-simply-laced case it consists of two roots (the highest long root and the highest short root). However, typically both lie on a wall of $K$.
Mar
27
comment Smoothing transverse intersections
If I understand correctly, what you want is not a resolution, but rather a smoothening.
Mar
26
comment How singular can the Stein factorization of a proper map between smooth varieties be?
@LaurentMoret-Bailly: To ensure that $X \to \hat{X}$ has connected fibers it is enough to take $\hat{X}$ to be normal. Then the first map has connected fibers and the second is finite, so it is the Stein factorization by the universal property. Of course, without normality of $\hat{X}$ this may be wrong.
Mar
26
comment How singular can the Stein factorization of a proper map between smooth varieties be?
@HNuer: Yes, of course, $X \to Y$ is only generically finite, but I guess this is precisely what was asked.
Mar
26
answered How singular can the Stein factorization of a proper map between smooth varieties be?