Questions tagged [affine-varieties]
Use this tag for questions related to an affine variety over an algebraically-closed field.
583
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Coordinate ring of an open affine variety?
I have trouble understanding what the coordinate ring of an open affine variety, i.e. the ring of regular functions on that variety, is (in the classical sense). Let $k$ be algebraically closed.
If $X ...
- 520
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Weight of a monomial
I have a question for the mathematicians in affine algebraic geometry: Given an algebraically closed field $k$, we define the projective $n$-space as the quotient space $\mathbb{P}^n = (k^{n+1} - \{0_{...
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Making clear the definition of 'affine variety' in Mumford's book.
I am reading "The Red Book of Varieties and Schemes" by Mumford. In section 4 the author defines the term affine variety:
An affine variety is a topological space $X$ plus a sheaf of $k$-...
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Understanding why set theoretic intersection is not necceraly a complete intersection
If it is true that for projective varieties one can show that:
$Z(f_1, f_2) = Z(f_1)\cap Z(f_2)$
for any homogenous polynomials, than why isn't true than any set theoretic complete intersection of ...
- 470
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rational map by F. Mangolte
I'm reading Real Algebraic Varieties by F. Mangolte.
Definition 1.3.22 (in the book)
If $X$ and $Y$ are algebraic varieties over a base field $K$ a rational
map $\phi:X\dashrightarrow Y$ is an ...
- 31
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Remark in Atiyah, Macdonald: non-singularity $\Rightarrow$ analytic irreducibility
In Atiyah, Macdonald, Introduction to Commutative Algebra, Chapter 11, p. 124, after Proposition 11.24, there is a
Remark. It follows from what we have said above that A is also an integral
domain. ...
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General way to determine whether a subset of a vector space is an affine space.
Given some implicit equations or a definition of a subset of a vector space, what is the requirement for it to define an affine subspace. I'm looking for something analogous to the test for vector ...
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Is $\textit{affine space}$ the same as $\textit{quotient space}$?
From the answer for this question, I understand that affine subspace is the same as affine subset, however (despite the somewhat misleading question's title), it doesn't say that affine space is the ...
- 607
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Quasi-affine variety with non finitely-generated global section
Let $U\subseteq\mathbb{A}^{3}$ be $V\left(xy\right)\setminus V\left(x,z\right)$. I'm trying to show that $U$, which is quasi-affine, has a non finitely-generated global section $\mathcal{O}_{U}\left(U\...
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Morphism of varieties is continuous between analytic varieties.
I'm seeking clarification on the significance of commutative diagrams in understanding the analytic topology of smooth varieties.
In Aleksander Horawa's notes, a commutative diagram is used to ...
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Fact checking: are these inclusion relations regarding algebraic varieties of polynomial ideals correct?
I'm studying inclusion identities within polynomial ideals theory. More precisely, i'm interested in the correspondece of an ideal $I\subseteq \mathbb{F}[\vec{x}]$ and its associated affine variety $\...
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Dominant rational maps and dimension of affine varieties
Say we have $f : X \dashrightarrow Y$ as a dominant rational map between two affine varieties. Is it necessarily true that $\dim Y \leq \dim X$?
$f$ being dominant means that we must have that $f(\...
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Existence of affine varieties and regular map solving functional equation for polynomials
I am wondering about the existence of projections from a multivariate polynomial $p\in\mathbb{R}[x_1,...x_n]$ to a polynomial $q\in\mathbb{R}[t]$ in the sense that there is a map $\phi:\mathbb{R}\to\...
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Zariski-density on almost diagonal embedding
It is not hard to see that the Gaussian integers $\mathbb{Z}[i]$ are Zariski-dense inside $\mathbb{C}$, seen as an affine space over $\mathbb{C}$. Consider now the set $$D = \{(z,\overline{z}) \in \...
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What are the conditions for a function $f\colon X \to Y$ to be regular when $X$ and $Y$ are prevarieties?
I'm currently following A. Gathmann's Algebraic Geometry, chapter 5 - varieties. I've seen the concept of prevarieties (ringed spaces with finite covers of affine varieties) and I'm stuck with the ...
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