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These are "type A quiver cycles" (a name chosen so as not to collide with type A quiver varieties, which involve taking quotients; here the quotient would be a point). Your guess for the closure is co …

First, one can resolve the Schubert varieties using Bott-Samelson manifolds, and discover that any two resolutions give the same class upon pushforward. (This good situation ends with K-theory, i.e. i …

What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P \hookrightarrow \mathbb P(V_\omega)$ where $V …

There are of course two moment maps to vary - the complex one and the real one. In most treatments of quiver varieties one fixes the complex level set to be zero and the real level set to a nonzero mu …

Pull back the equation $[W] = [H]^{codim\ W} \in \mathbb P^N$. This cohomological pullback is computable set-theoretically, as $[W\cap G]$, if the intersection is transverse.

By "coadjoint orbits" I assume you mean "of a compact Lie group, but then endowed with invariant complex structures". In which case the answer is no. There is a flat family whose general fiber is $\ma …

This non-answer grew too long for a comment. Let's change coordinates, $B_i = \check\rho(t)A_i$, where $\check\rho(z)$ is the diagonal matrix $diag(t,t^2,t^3,\ldots,t^n)$. Then the original equations …

Yes. I assume that your $M_n$ is what is more usually denoted $\overline{M_{0,n}}$. Then the answer is yes, there is a natural map $\overline{M_{0,n}} \twoheadrightarrow M_L$, for each $L$. Specifica …

In general if $T$ acts on a projective variety $X$ with moment polytope $\Phi(X)$, then a general point $x\in X$ will have $\Phi(\overline{T\cdot x}) = \Phi(X)$ i.e. be an abnormal toric variety with …

See them as pushforwards of line bundles from the flag manifold by using Borel-Weil fiberwise. Then use Borel-Weil up on the flag manifold. Jerzy Weyman's book is the source I know of for these techni …

For a toric variety, such as this cone over the 3rd Veronese of $\mathbb P^1$, the complement of the open torus orbit is an anticanonical divisor (indeed, that is the one given by the unique $T$-invar …

Disclaimer: I am far from expert in the topics under discussion. Let $A \subseteq V$ be a closed affine cone in a vector space, defined by polynomial equations. In classical "projective duality", on …

It's not true scheme-theoretically, at least. Let $X=\mathbb P^4$ with coordinates $w,x,y,z,\Omega$, let $A$ be the $xy\Omega$-plane, and $B$ the union of the $wx\Omega$- and $yz\Omega$-planes. Then $ …

I think you're absolutely right that the function $(i\in \mathbb N)\mapsto $interestingness($H^i$) is a rapidly decreasing function. I heard that Gel$'$fand compared it to the successive derivatives o …

In general, $V_{k\rho} \cong \bigotimes\limits_{\beta\in \Delta_+} (\mathbb C_{k\beta/2} \oplus \mathbb C_{(k-2)\beta/2} \oplus \ldots \oplus \mathbb C_{-k\beta/2})$ as $T$-representations, provable v …