# Questions tagged [zariski-topology]

For questions about the topology of schemes and (classical) algebraic varieties.

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### Problem 6.8 in Fulton's Algebraic Curves: the nonvanishing locus of a regular function is open

Question: Let $U$ be an open subset of a variety $V$, $z\in k(V)$. Suppose $z\in O_p(V)$ for all $P\in U$. Show that $U_z = \lbrace P\in U \mid z(P) \ne 0 \rbrace$ is open. Attempt: Initially I want ...
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### Proving certain subset of the product of the affine line and the Grassmannian is closed in the Zariski topology

Let $k$ be an algebraically closed field of characteristic $0$, $0 < d < n$ be integers,$\mathrm{Gr}_k(d,n)$ be the Grassmannian parametrizing all linear subvarieties of dimension $d$ contained ...
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### Closure of quasi-projective variety: if $X=V(I)\setminus V(J)$, must $\overline{X}=V(I)$?

Let $X=V(I)\setminus V(J)$ in a complex projective space $\Bbb P^n$, where $I,J$ are ideals of complex polynomials in $n+1$ variables and $V(\ldots)$ their common zeros. I mean $X$ is locally closed ...
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### Zariski topology on $\mathbb{Z}$

I'm trying to understand the Zariski topology on $\text{Spec}(\mathbb{Z})$. I've just learned about this new concept and I wanted to compute this topology for a more concrete example to see how it ...
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### Image of a projective variety is a projective variety

I am using Karen E. Smith et al.'s Invitation to Algebraic Geometry, and was wondering the following: if $$\phi : V \subseteq \mathbb{P}^n \to W \subseteq \mathbb{P}^m$$ is a morphism of ...
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### Is open always Zariski open in the context of Algebraic geometry, unless otherwise mentioned?

Going through Smith et al.'s Invitation to Algebraic Geometry I sometimes find myself wondering whenever they use the word open, do they mean open in the usual sense on $\mathbb{A}^1$ for example, or ...
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