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As pointed out, the statement is false as written. To get a true statement, you must assume also that $f$ is square-free (or equivalently that the scheme $V(f)$ is reduced). Then, if you assume that …

Michel Brion's Lectures on the Geometry of Flag Varieties answers all of your questions in the special case $G = SL_n$ and $P = B$ (the Borel subgroup of upper triangular matrices). See Section 1.4. …

First, whether a class is pseudoeffective or not depends only on its numerical equivalence class. (The pseudoeffective cone is the closure of the cone of big classes, and $D$ is big if and only if $n …

Several years ago, I participated in a learning seminar in tropical algebraic geometry and collected several helpful survey articles. (This was before Maclagan and Sturmfels' book was written, which …

You might be better off studying an analytic approach to algebraic geometry first. The classical reference for this approach to algebraic geometry is Griffith and Harris' Principles of Algebraic Geom …

I am looking for generalizations of the following construction. Let $G$ be a connected, reductive group and let $\theta : G \rightarrow G$ be an involution. Let $H = G^{\theta}$ be the subgroup of $ …

On your third blowup, you need to check for singularities at all points that lie above your singular point, not just the point with local coordinates (0,0). Fortunately, the equations you get from se …

(This should be a comment but I don't have enough reputation to leave one.) The quotient of $U$ by $PGL(2)$ is $\mathbb{P}^1 \backslash \{ 0, 1, \infty \}$. To get something compact, you need to all …

Malheureusement, this is not true, not even for the weak order. This can be seen for example when $G = GL(4)$ and $K = GL(2) \times GL(2)$. Then $K \backslash G / B$ is parameterized by involutions …

A good reference is Neukirch's Algebraic Number Theory. It takes the analogy between algebraic number theory and algebraic geometry seriously, and it includes coverage of class field theory. But you'd …

Let $V$ be a complex vector space of even dimension. Then the homogeneous space $SL_{2n}(V) / SP_{2n}(V)$ is known to parametrize the space of non-degenerate skew-symmetric bilinear forms on $V$. (1 …

If you are interested in intersection theory of varieties with $G$-actions, then you want to study equivariant intersection theory. This theory exploits the $G$-action in a way that leads to deeper i …

The numbers you refer to are known as Kostka numbers. They are discussed in standard references like Fulton's Young Tableaux and Stanley's Enumerative Combinatorics. The weights of a tableaux are of …

(1) Assuming you are referring to the coefficient field for your cohomology theory, then yes, the result immediately extends to $\mathbb{R}$ and $\mathbb{C}$ coefficients. (2) Two sources are Fulton' …

Some additional resources, which are more on the algebraic side than the symplectic side: 1) Bill Fulton's Eilenberg lectures on Equivariant Cohomology in Algebraic Geometry, available at David Ander …