Questions tagged [valuative-criteria]

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Direct proof that for $X$ an integral scheme with every valuation on $K(X)$ having unique center, the same is true for all closed integral subschemes

Suppose $X$ is an integral scheme of finite type over a field $k$, and assume that every valuation on $K(X)/k$ has a unique center. Is there a direct proof that the same is true for every integral ...
5
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2answers
764 views

When are valuative criteria useful?

We have valuative criteria for properness and universal closedness of morphisms of schemes. I would agree that these criteria shed some light on the geometric nature of these morphisms. However, are ...
5
votes
1answer
416 views

Valuative criterion to extend morphism of schemes

Let $k$ be an algebraically closed field, $X, Y$ integral $k$-schemes and $Y$ proper over $k$. Let $U$ be a non-empty open subset $U \subset X$ and $f:U \to Y$ a morphism of finite-type. Suppose ...
22
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3answers
729 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 ...
14
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1answer
675 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
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1answer
1k 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 $K$-...
9
votes
1answer
849 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
3answers
1k 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 Spec(K)→...
9
votes
1answer
2k 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) ...