Hasse diagrams of G/P_1 and G/P_2

in the Paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.5052&rep=rep1&type=pdf at the end, we can see Hasse diagrams for several projective, homogeneous G-varieties for G being a exceptional linear algebraic group. Note that D4/P1 is isomorphic to a six dimensional quadric, that i will denote as Q6. In an unfinished book by Gille, Petrov,N. Semenov and Zainoulline, which can be found … Read more

Unibranch partial normalization

In a paper I recently read something about the “unibranch partial normalization” of a curve. Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it possible to find a minimal ring $R^u$ between $R$ and its normalization $R’$ in $K$ such that all localizations of $R^u$ in the … Read more

Modules over an Azumaya algebra and modules over the associated Brauer-Severi variety

Assume A is an Azumaya algebra of rank r2 on a smooth projective scheme Y over C. Let f:X→Y be the Brauer-Severi variety associated to A. I read here in a comment that the category of modules over A is equivalent to the categroy of modules on X which restrict to every fiber as a … Read more

Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let X be a scheme of finite type over C and let Z↪X be a closed subscheme. Consider the associated closed inclusion Zan↪Xan between their analytifications (regarded as topological spaces). Is this a strong neighborhood deformation retract? By this I mean, can I find for every neighborhood U of a point z in Zan another … Read more

Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let G be a finite group,let Rep(G) denote the category of finite dimensional representations over C. Let V,W be representations of G in Rep(G). One can define a bilinear form on Rep(G) or inner product in K0(Rep(G)) (in Teleman’s notes) as dimCHom(V,W)G which is G invariant of Hom(V,W).Then there … Read more

Rational connectedness of certain subvarieties of the linear series

Let X be a smooth projective hypersurface in P3, |OX(a)| be the complete linear system for some integer a>0. Ofcourse, a general element of the linear system is a smooth curve. Denote by V the subvariety of |OX(a)| parametrizing reducible curves (i.e., with at least 2 irreducible components). Is any irreducible component of V rationally … Read more

Why is the Hodge Conjecture so important?

The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I’m able to understand what it says, and at first glance I do find it a very nice assertion, I cannot grasp yet why it is … Read more

Dimension of the singular locus of $\mathcal M_X(r,d)$

Let $X$ be an algebraic smooth curve of genus $g$ over $\mathbb C$, and let $\mathcal M_X(r,d)$ (resp $\mathcal M_X^0(r)$) be the moduli space of vector bundle of rank $r$ and degree $d$ (resp. with trivial determinant) over $X$, we know that when $g=r=2$ and $d\equiv 0 \mod 2$ this space is smooth. So we … Read more