# Questions tagged [affine-geometry]

For questions about algebraic geometry that focus on affine space. For affine mappings in linear algebra (i.e. linear mappings plus translations), please use the linear-algebra tag or another appropriate tag.

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### Prove continuity of the affine extension mapping between geometric simplicial complexes

Let $\Delta_1$ and $\Delta_2$ be geometric simplicial complexes. Let $K_1$ and $K_2$ be their associated abstract simplicial complexes. Let $f: V(K_1) \to V(K_2)$ be a simplicial mapping. We define ...
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### Affine Map as a Morphism of Affine Vector Spaces

I've recently took interest in morphism and category theory and I'm amazed how it offers a very general notion. However, I'm struggling to apply this for the affine vector spaces. I've seen that a ...
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### A quadric has the origin as centre if and only if its equation has no terms of degree $1$ [closed]

I'd like to know how to prove that if a(n affine) quadric of A^n (regarded as the affine space canonically associated to K^n, being a field) has the origin as centre, then its equation has no terms of ...
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### Does unique convex combination imply affine independence?

Why we need affine independent to ensure the unique representation of a vector from convex hull. As far as I understood, the converse of the theorem is not true. In other words, if any vector 𝑣 is in ...
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### Hausdorff-ness of irreducible affine algebraic sets [closed]

In what case is $X$, an irreducible affine algebraic set not a Hausdorff topological space with respect to the Zariski topology? My understanding is that this would only occur if $X$ is infinite, ...
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### Prove $S = \{ (a,b) \in \mathbb{A}^2 \ | \ a \overline{a} + b \overline{b} = 1\}$ is not Zariski closed

I'm working on a problem where I have to prove that Consider the subset $S$ of $\mathbb{A}^2$, the affine space over $\mathbb{C}$, defined by  S = \{ (a,b) \in \mathbb{A}^2 \mid a \overline{a} + ...
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### Projective closure as a closure operator

I am studying basic projective and affine geometry and I've come across the notion of projective closure (by now, only in spaces obtained from K^n by projectivisation). It has been defined (somewhat ...
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### Characteristic polynomial and bounded regions.

I know that the number of bounded regions of a homogeneous hyperplane arrangement $\mathcal{A}$(a collection of n hyperplanes) in $\mathbb R^d$is ${ n - 1 \choose d}$ but how can this be expressed in ...
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### Understanding contraction in hyperplane arrangements.

Here are two figures that shows a hyperplane arrangement's contraction (this is from McNulty book, "Matroids, a geometric introduction"): I am not sure why a became a line in the right ...
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### If $T:\ \mathbb R^n\longrightarrow\mathbb R^n$ is additive then $T$ is linear

Let the bijection $T:\, \mathbb R^n\longrightarrow\mathbb R^n,\ n\geq2$ satisfy \begin{align} T(u+v)=T(u)+T(v),\ \forall u,v\in\mathbb R^n\tag1 \end{align} and \begin{align} T\big(\langle u\rangle\big)...
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