# Questions tagged [zariski-topology]

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### Defining path on the prime spectrum

If $p$ and $q$ are two prime ideals of a commutative ring $R$ such that $p \subseteq q$, then we can easily define a continuous function (a path) $f$ from the unit interval $[0,1]$ to the prime ...
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### Symmetric products of varieties and projective bundles

Given a smooth projective geometrically connected curve $C$, a symmetric product of $C$ has the structure of a projective bundle over the Jacobian of $C$ (e.g. see Symmetric powers of a curve = ...
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### Locally affine varieties and du Val singularities

Let me start with an apologetic disclaimer: I am very far from an algebraic geometer, so this question might be crudely formulated. I have a specific question about du Val singularities, but while ...
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### The Grothendieck ring of varieties with classical Zariski

Consider $\mathcal{V}_k$, the category of $k$-varieties over the finite field $k = \mathbb{F}_q$ with $q$ elements. We see varieties in the old "classical" sense of the word, foreseen with the old ...
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### On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian

All rings below are commutative with unity. If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...
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### A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...
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### Convergent sequences in projective varieties

It's very well known that if $X$ is an irreducible projective variety (feel free to assume that the base-field is $\mathbb{C}$), then any two points $x,y\in X$ can be connected by the image of a non-...