# Questions tagged [affine-geometry]

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### Almost blocking sets in $\mathbb F_q^2$

$\newcommand{\F}{{\mathbb F}}$ Let $q$ be an odd prime power. A blocking set in the affine plane $\F_q^2$ is a set blocking (meeting) every line. A union of two non-parallel lines is a blocking set ...
258 views

### Status of an open question in Artin's “Geometric Algebra”

In Artin's book "Geometric Algebra", Chapter II, the author states some axioms for geometry (section 1) and then begins to prove some results about the symmetries of the geometry (section 2). The ...
227 views

### Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that ...
85 views

### Barycenters and the axiomatic of affine geometry

There are several ways to define an affine space, either by starting from a transitive action of a vector space on a set of points, or listing sets of axioms related to parallelism in the spirit of ...
217 views

### Parallel Ricci condition - Status report and bibliography

First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's ...
121 views

### Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex if its restriction to each line is. An affine ...
651 views

### Theorems in affine geometry which can be proved using only dilations, generalized to metric spaces

Background: significant parts of E. Artin, Geometric algebra, Wiley-Interscience, New York, 1957 can be seen as consequences of the statement (1) "the inverse semigroup generated by dilations in ...
111 views

### Flatness of modules over dual numbers

Let $X$ be a smooth, affine complex surface, and $M$ be a coherent $\mathcal{O}_X$-module. Denote by $D:=\mbox{Spec}(\mathbb{C}[t]/(t^2))$ and $X_D:=X \times_{\mathbb{C}} D$, the trivial deformation ...
160 views

### Reference for “topological affine spaces”

I am wondering if there is a topological version of affine spaces as a topological space along with a free transitive (continuous) action of a topological vector space on it? Here is a notion so-...
114 views

Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[... 0answers 116 views ### Factorially closed, finitely generated$k$-sub-algebra of$k[X_1,X_2,X_3]$, where$k$is algebraically closed field of positive characteristic Let$S$be a sub-ring of a commutative ring with unity$R$. Then$S$is called factorially closed in$R$if$a,b \in R$and$ab \in S\setminus \{0\} \implies a,b \in S$. Let$k$be an algebraically ... 0answers 71 views ### If a partial normalisation of an affine variety is affine? Let$X$be an affine variety. Let$x$be an isolated singular point of$X$. Let$U$is an affine neighbourhood of$x$such that$U\setminus \{x\}$is smooth. Let$\pi:\tilde{U}\to U$be the ... 0answers 80 views ### Characterizing normal vectors of affinely independent subsets of the hypercube Suppose we are looking at the hypercube in$\mathbb{R}^n$. I tend to the think of the cube with vertex coordinates 0 or 1, but maybe this is easier for the$\pm1$cube. Now suppose we have an affine ... 0answers 116 views ### What is this 2-form on a Lagrangian torus fibration? Suppose we are given a regular$2n$-dimensional Lagrangian fibration$\pi : (M,\omega) \to B$with connected, compact fibers. Then it is well-known (Arnold-Liouville) that each fibre is a Lagrangian ... 0answers 115 views ### Affine-regular hexagon in convex body An affine-regular$n$-gon is a non-degenerate affine image of the regular$n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set$K\subseteq \mathbb R^2$... 0answers 139 views ### A questions related to the Markus conjecture for special affine manifolds An affine manifold$M$is called special if there is a parallel volume form$\omega$on$M$, and a nowhere vanishing vector field$\mathcal{V}.$Here we need to point out that any affine manifold of ... 0answers 620 views ### Does the coordinate ring of affine variety admit a structure of infinite dimensional variety? We work in the category of algebraic varieties over some algebraically closed field$k$. By infinite dimensional variety I mean a filtration: $$V_0\subset V_1\subset V_2\subset\ldots$$ where each$...
Say you have an affine variety $X$ in $n$-dimensional affine space. (You can even assume we are over $\mathbb{C}$, but I believe the nature of my question is algebraic). I want to bound from above ...
Is there some nice compactification of the affine space with a del Pezzo surface of degree $\le 4$ (for example with a cubic surface) ? More precisely, I would like a projective algebraic variety $X$ ...