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20
votes
3answers
536 views

Can continuity of a function be checked by restricting to smooth curves?

Well-known example: Consider the function $$f(x,y)=\left\{\begin{array}{c} \frac{x^2y}{x^4+y^2} & \text{if }(x,y)\neq(0,0)\\ 0 & \text{if }(x,y)=(0,0) \end{array}\right.$$ When restricted to ...
11
votes
1answer
409 views

Can a single DVR witness all specializations on a variety?

If $X$ is a noetherian scheme with points $x$ and $\xi$ so that $x$ is in the closure of $\{\xi\}$, then there exists a discrete valuation ring $V$ and a map $Spec(V)\to X$ sending the generic point ...
7
votes
1answer
644 views

Valuative criterion for properness

Let $f : X \rightarrow Y$ be a finite type morphism of Noetherian schemes. The valuative criterion for properness runs as follows. Suppose that for any DVR $R$ with fraction field $K$ that any ...
6
votes
1answer
477 views

Example where you *need* non-DVRs in the valuative criteria

The valuative criterion for separatedness (resp. properness) says that a morphism of schemes (resp. a quasi-compact morphism of schemes) f:X→Y is separated (resp. proper) if and only if it ...
5
votes
2answers
477 views

Can the valuative criteria be checked “on a dense open”?

The valuative criterion for separatedness (resp. properness) says that a noetherian scheme X is separated (resp. proper) if and only if for any DVR R, with fraction field K, any map ...
3
votes
1answer
708 views

Can the valuative criteria for separatedness/properness be checked “formally”?

Suppose f:X→Y is a morphism of finite type between locally noetherian schemes. The valuative criterion for separatedness (resp. properness) says roughly that f is a separated (resp. proper) ...