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Oct
21
awarded  Yearling
Oct
21
awarded  Yearling
Oct
21
awarded  Yearling
Aug
17
awarded  Nice Answer
Oct
21
awarded  Yearling
Jul
12
comment Is the Springer resolution a blow-up?
In the simplest case of $\mathfrak{sl}_2$, the Springer resolution is easily seen to be the blow-up at the origin. In general, however, the Springer resolution is not a blow-up.
Mar
6
comment The `set' of all principal G bundles over `all' spaces
Within the context of stacks in algebraic geometry, $BG$ (or $pt/G$) is absolutely the notation that would be used. I suppose you don't like this because it is also the notation for the classifying space of $G$? From my perspective, I don't particularly like the notation $Bun(G)$, only because $Bun_G(X)$ is typically used to denote the moduli of $G$-bundles on the space $X$. But maybe others disagree. I'm struggling to think of a good alternative though.
Feb
12
answered A^1-invariant for Vector Bundles?
Feb
11
comment What is the dimension of a sheaf?
The sheaf in question is a sheaf over the residue field, which means that it can be viewed as vector space over the residue field.
Jan
23
answered Simple question in the representation of SL(2,C)
Jan
5
comment $(\mathfrak{g},K)$-modules and parabolic category $\mathcal{O}$
A general reference for category $\mathcal{O}$ that I really like: Humphreys' "Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$"
Nov
19
comment Classification of certain algebraic curves
One more comment: If you are simply looking for curves which possess a line bundle satisfying the equality, then any curve possessing a $g^1_4$ obviously works. This includes all hyperelliptic curves, but also all curves of genus at most $6$. More generally, you might try checking when the Brill-Noether number associated to $d=deg(L)$ and $r=\frac{1}{2}deg(L)-1$ is non-negative. I haven't done the calculation, but it seems that using this strategy you might find that every curve possesses a line bundle satisfying your equality?
Nov
19
comment Classification of certain algebraic curves
This questions seems unclear at the moment. First of all, every curve has a special line bundle satisfying the equality $h^0(L) = \frac{1}{2}deg(L)+1$; namely the canonical bundle. Hyperelliptic curves are unique because they also possess a line bundle which satisfies the equality which is neither the trivial bundle nor the canonical bundle. So, is your question asking which curves possess some special line bundle satisfying $h^0(L) = \frac{1}{2}deg(L)$? Or, are you asking for a list of all line bundles on all curves which satisfy the equality, akin to Clifford's theorem?
Nov
9
comment real orbits of highest weight vectors
Regarding question 1: Fix a Borel subgroup $B$ and corresponding highest weight vector $v_{\lambda}$. Then for any other Borel subgroup $B'$, there exists $g \in G$ such that $gBg^{-1} = B'$. In this case, the highest weight vector $v_{\lambda}'$ associated to $B'$ is simply $gv_{\lambda}$. Therefore $G$ does act transitively on highest weight vectors. It seems to me that the stabilizer would then be the unipotent radical of $B$.
Nov
9
awarded  Good Answer
Nov
8
comment Is base affine space a trivial fibration?
Presumably the base affine space is $G/U$ (I haven't heard the terminology 'base affine space' before)? I think it's helpful to consider the case where $G=SL_2$, so that the flag variety is $\mathbb{P}^1$ and $G/U$ is the complement of the origin in $\mathbb{A}^2$, which isn't a trivial $\mathbb{G}_m$-bundle over $\mathbb{P}^1$.
Nov
8
comment Deformation of Line bundles over dual numbers
A short exact sequence $0 \to L \to M \to L \to 0$ enables you to give $M$ the structure of a module over $X'$. You just need to be able to say how $\epsilon$ acts on $M$. Composing the projection $M \to L$ with the inclusion $L \to M$ tells you how $\epsilon$ acts.
Oct
22
awarded  Yearling
Oct
19
awarded  Autobiographer
Sep
27
comment Which bundles does the character variety parameterize?
When $G = U(n)$, the character variety parameterizes vector bundles over $C$. This is the Narasimhan-Seshadri Theorem. When $G = GL_n$, the character variety parameterizes Higgs bundles. This is part of what's known as non-Abelian Hodge theory.