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We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.

How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber at $0$?

In particular, how should the moment map images of the one-dimensional $T-$orbits change?

Any suitable references? Thanks!

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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 the multigraded Hilbert polynomial, it too is constant. Then the moment polytope is the support of the DH measure, so it, too, is constant.

Here's an example to consider: $F = Proj\ \mathbb C[x^1,y^1,z^1,\epsilon^0]/\langle xz - \epsilon y^2\rangle$ over $Spec\ \mathbb C[\epsilon]$, where the exponents indicate the degree, and $\mathbb C^\times$ acts on $x,y,z,\epsilon$ with weights $0,1,2,0$ respectively. Then for $\epsilon\neq 0$, the $\epsilon$-fiber is a smooth plane conic with moment polytope $[0,2]$ (bearing DH measure = Lebesgue measure). But for $\epsilon = 0$, it's a union of two lines, whose moment polytopes are individually $[0,1],[1,2]$. In particular the general fiber has two fixed points but the special fiber has three.

What's going to happen to the $1$-d orbits is that they may break (as in the example), and new ones may form (similar to the extra fixed point in the above example).

If as the question title indicates, you really want references about toric varieties, the canonical one is Bernd Sturmfels' 1990 paper on Gr\"obner bases of toric varieties.

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  • $\begingroup$ In your example, the generic fiber is smooth. When the generic fiber is singular, would the moment polytope always stay constant in a flat degeneration over $\mathbb{A}^1$? $\endgroup$
    – Qiao
    Commented Jun 2, 2015 at 17:30
  • $\begingroup$ Yeah, smoothness is irrelevant. If $T$ acts on a positively graded ring $R = \bigoplus_n R_n$ hence on $Proj\ R$, the moment polytope of $Proj\ R$ is the closure of its rational points $\{\lambda \in {\mathfrak t}^*\ :\ n\lambda$ is a $T$-weight of $R_n$ for $n \gg 0\}$. This doesn't use smoothness, characteristic $0$, whatever. (But it may not be convex if $R$ isn't a domain, e.g. if $Proj\ R$ isn't connected.) I suppose the base ring should be an algebraically closed field so that the $T$-irreps will be classified by the weight lattice. $\endgroup$
    – Allen Knutson
    Commented Jun 5, 2015 at 13:29

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