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4h

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2d

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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

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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

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Does a fully faithful functor between triangulated categories induce embedding of their Grothendieck groups?
Density is a very strong assumption! 
2d

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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

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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 
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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 
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What is the applications of the dgenhancements 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 
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Connected vs Irreducible Subvarieties
Sometimes smoothness is easier to check than normality, and of course it also implies irreducibility of connected components. 
Mar 28 
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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 
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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 
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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 
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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 
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Root in positive Weyl chamber
It is easy to classify $R \cap \overline{K}$. In the simplylaced case this set consists just of one root (the highest root). In the nonsimplylaced 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 
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Smoothing transverse intersections
If I understand correctly, what you want is not a resolution, but rather a smoothening. 
Mar 26 
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How singular can the Stein factorization of a proper map between smooth varieties be?
@LaurentMoretBailly: 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 
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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? 