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While working with flat degenerations of flag varieties and Schubert varieties I've noticed that among the numerous known constructions there doesn't seem to be a single one that doesn't turn out to b …

Consider the Lie group $G=SL_n(\mathbb C)$ with Borel subgroup $B$ and maximal torus $T\subset B$. I'm interested in the (Zariski) closures of $T$-orbits in the flag variety $F=G/B$. Now, as far as I …

Didn't get a single comment in over a day at math.SE, so maybe the question is more appropriate here. I'm looking for a reference to a generalization of Hilbert's Nullstellensatz to the multiprojectiv …

To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $ …

Consider a convex rational polyhedral cone $C\subset\mathbb R^m$ with vertex at the origin. Let $X$ be the corresponding affine toric variety, i.e. $\mathbb C[X]=\mathbb C[\mathbb Z^m\cap C^\circ]$. N …

For $n\ge 2$ consider the set of Plücker variables $X_{i_1,\dots,i_k}$ with $1\le k\le n-1$ and $1\le i_1<\dots<i_k\le n$ and the ring $R$ of polynomials in these variables (with complex coefficients) …

I'd say that the relevant fact here is as follows. For two Borels $B_1$ and $B_2$ with a common maximal torus $T$ let $x_1$ be the unique $T$-fixed point in the open $B_1$-orbit. Then the $B_2$-orbit …

This is essentially an extension of my comment, just to answer the actual "is this the only case?" question. It is not, $Z$ will be affine whenever $S$ is an antichain in the Bruhat order. Indeed, thi …

Consider the Plücker embedding of the variety of complete flags in $\mathbb C^n$: $$F_n\subset\mathbb P(\bigwedge\nolimits^1\mathbb C^n)\times\dots\times\mathbb P(\bigwedge\nolimits^{n-1}\mathbb C^n). …

First of all, as mentioned in the comments, multiprojective toric varieties standardly arise from Minkowski sums rather than unions. Let me phrase this in the context of toric varieties of lattice poi …

Consider a polynomial ring $\mathbb C[x_1,\ldots,x_n]$ that is $\mathbb Z_{\ge 0}$-graded by degree. Let $I$ and $J$ be two homogeneous ideals therein with the same Hilbert series, i.e. with their hom …

Victor Petrov essentially answered your question showing that this projective embedding is, in general, not minimal. I'll just try to explain why this other embedding is, in fact, minimal by dimension …