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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 …

(Building on Pedro Montero's comment.) I don't know much about studying general such morphisms, but toric morphisms are easy to think about. Four general points is $PGL(4)$-equivalent to the four $T$ …

I assume you mean that $T$ acts preserving each fiber. Then the flatness says that the multigraded Hilbert polynomial is constant. As the Duistermaat-Heckman measure is the leading-order behavior of t …

It's not. These are constructible as ${\mathbb A}^2/\Gamma$ for $\Gamma$ a finite subgroup of $SL(2)$, and you can recover $\Gamma$ as $\pi_1$ of a punctured neighborhood of the singularity. If you co …

I think I'd want to deal with partially defined functions $f: [n] \to [n']$, with the property that if $F$ is a face of $\Delta$, then $f(F)$ is a face of $\Delta'$. The linear extension of such an $f …

The four new $P^1$s have self-intersection $-1$. The four old $P^1$s (or rather, their proper transforms) have self-intersection $-2$. So there's no automorphism that's going to switch the two sets o …

Since you already know how to blow up along either ${\mathbb P}^1$ individually, we can concentrate on what's happening nearby the intersection. Which means we can work affinely. Then the polyhedron …

A codimension $d$ face of a polytope is called rationally smooth if it lies on only $d$ facets, because this is exactly the condition for the corresponding toric variety to have only orbifold singular …