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I think this has a sociological, not mathematical, answer. Whenever you have a structure that could be chosen on a mathematical object, it is handy to have one adjective to say you've made this choic …

(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 $T$ …

In general for a Grassmannian $Gr_k(\mathbb C^n)$, the Schubert polynomials are Schur polynomials in $k$ variables, one for each partition $\lambda$ in a $k\times (n-k)$ rectangle. In this case $S_\la …

This is more reasonable if you insist that $C$ have a given Hilbert polynomial. Otherwise, consider the case $A = \emptyset$, $B = \bf P = \bf P^2$. Then you have curves of every degree, so your $V(A, …

There are many GIT quotients, since to define one requires a choice of $G$-line bundle, so a pair of naturals for each $F$. There's an obvious democratic choice -- $(a,b) = (1,1)$ for every $F$ -- b …