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11
votes
1answer
1k views

When is fiber dimension upper semi-continuous?

Suppose f:*X*→Y is a morphism of schemes. We can define a function on the underlying topological space Y by sending y∈Y to the dimension of the fiber of f over y. When is this function upper ...
4
votes
2answers
226 views

Understanding the definition of the quotient stack $[X/G]$

I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles. Explicitly, let $G$ be an affine smooth group ...
2
votes
2answers
217 views

Are orbits of an affine algebraic monoid affine?

Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let ...