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Yes, there is a formula for this due to Brion (it's Lemma 2 in his paper Multiplicity-Free Subvarieties of Flag Varieties). First, for any $B$-equivariant coherent sheaf $\mathcal F$ on $X$, there is …

Well, kind of -- a line bundle is nef if and only if its tensor product with any ample line bundle is ample.

Let $X$ be a scheme over an algebraically closed field $k$ of positive characteristic $p$. Recall that the absolute Frobenius morphism $F : X \to X$ is the map which is the identity on points and the …

A weakening of 2) is an exercise (III.12.6) in Hartshorne: Let $X$ be an integral projective scheme over an algebraically closed field $k$ and assume that $H^1(X, \mathcal O_X) = 0$. Let $T$ be a conn …

Let $S$ be a graded ring with $A := S_0$. Set $X := \textrm{Proj} (S)$. Then the projective coordinate ring of $X \times_A X$ is the graded ring $ \bigoplus_{n \geq 0} S_n \otimes_A S_n $, cf Hartsho …

This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and daunti …

This is a continuation of various questions about Chevalley groups over rings, cf these two questions (and a rather bad question of mine here). Consider a semisimple Lie algebra $\mathfrak g$ over $\m …

Fix an algebraically closed field $k$ of arbitrary characteristic $p$ and let $R$ be a finite-dimensional local $k$-algebra (so in particular $R$ is Artinian and Noetherian). Let $S_n$ be the symmetri …

For question 1, let $L_\alpha$ denote the Levi of the parabolic $P_\alpha$; then $L_\alpha$ has derived group $SL_2$. Let $B_\alpha$ denote the Borel of $L_\alpha$ such that $B \cap L_\alpha = B_\alph …

This statement is false in general for algebraic groups. It's true in characteristic 0, but it is not in general true in positive characteristic. Instead, one has a weaker statement in positive charac …

As Vladimir mentions, the statement of your theorem is unclear. However, since you have written the $G$- and $H$-actions as tensor product actions where each acts nontrivially on one tensor factor and …

The reason your argument doesn't work is because it's not true that $\text{Sym}^l \mathfrak n^\vee$ has a filtration with 1-dimensional graded pieces where $T$ acts with anti-dominant weights. In fact …

BACKGROUND Let $X$ be a variety over an algebraically closed field $k$ of positive characteristic $p$. Let $F : X \to X$ denote the absolute Frobenius morphism, i.e. the morphism that is the identity …

NOTE: I am very much not an expert on affine Kac-Moody groups or ind-varieties, so the following may be vague. The first question is: Has anyone developed a theory of Frobenius splitting for ind-vari …

Let $X$ be a smooth variety and consider the diagonal $\Delta \subseteq X \times X$. It seems to be well-known that the exceptional divisor in the blow-up of $X \times X$ along $\Delta$ is isomorphic …