All Questions
Tagged with ag.algebraic-geometry mg.metric-geometry
86 questions
102
votes
6
answers
11k
views
Is there an analogue of curvature in algebraic geometry?
I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...
- 29.2k
96
votes
4
answers
5k
views
A curious relation between angles and lengths of edges of a tetrahedron
Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...
- 4,316
25
votes
2
answers
1k
views
Geometry of complex elliptic curves
Is there an elliptic curve in CP^2 whose induced Remannian metric ( induced from the Fubini-Sudy metric on CP^2) is Euclidian flat?
- 1,387
23
votes
12
answers
15k
views
Textbook for undergraduate course in geometry
I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (...
21
votes
1
answer
975
views
Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points
The following question resisted attacks at Math SE, so I thought I would try posting it here.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least ...
- 3,527
20
votes
5
answers
1k
views
Historical use of figures in geometry
I was surprised to learn from John Stillwell's comment in answer to the
question,
"Can the unsolvability of quintics be seen in the geometry of the icosahedron?",
that
There is not a single picture ...
- 151k
20
votes
5
answers
1k
views
From convex polytopes to toric varieties: the constructions of Davis and Januszkiewicz
One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
- 24.7k
18
votes
2
answers
2k
views
Which platonic solids can form a topological torus?
8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the ...
- 453
18
votes
2
answers
700
views
Can all unit-distance graphs have their vertices at algebraic integers?
A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$.
Obviously, we can ...
- 12.4k
17
votes
4
answers
772
views
Partitions of $\mathbb{R}^d$ by implicit polynomial equations
Given a polynomial
$p(x_1,x_2,\ldots,x_d)$
in $d$ variables, with maximum degree $k$,
what is the maximum number of
components of $\mathbb{R}^d$ minus $p(\ldots)=0$?
In other words, into how many ...
- 151k
16
votes
2
answers
1k
views
Algebraic surface of constant width?
Does there exist an irreducible polynomial $f \in \mathbb{R}[x, y, z]$ such that:
$$ V := \{ (x, y, z) \in \mathbb{R}^3 : f(x, y, z) \leq 0 \} $$
is a solid of constant width with a finite symmetry ...
- 12.4k
16
votes
2
answers
1k
views
Integer lattice points on a hypersphere
Is the following statement true?
For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice points ...
- 513
16
votes
1
answer
774
views
Minimizing the excursion of a sum of unit vectors
I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose
sum is zero:
$$ v_1 + v_2 + \cdots + v_n = 0 \; .$$
Now I form the closed polygon $P$ in space by placing them head to tail.
So the ...
- 151k
15
votes
1
answer
669
views
Affine "real algebraic geometry" of hyperbolic space?
Real algebraic geometry, at least to start with, traditionally studies the zero-sets of real polynomials in a given set of variables. But treating, say, the Euclidean plane as an uncoordinatized ...
- 17.6k
15
votes
1
answer
846
views
What is the longest algebraic curve?
Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$.
Is $L(d)\leq d P(E)/2$, where $P(E)$ is the ...
- 7,836
12
votes
1
answer
559
views
Square lying on moving chord of a simple closed curve
Consider a simple closed curve $C$ in $\mathbb{R}^2$. For any points $a$ and $b$ on this curve we associate a point $c_1$ on the left and $c_2$ on the right side to the chord ab, such that $ac_1bc_2$ ...
- 415
11
votes
2
answers
2k
views
Non-Kahler Complex manifolds
For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define ...
11
votes
3
answers
1k
views
What is the minimum-curvature curve interpolating a given set of points in the plane?
We are given a set $X$ of $n\ge 3$ points in $\mathbb{R}^2$, belonging to the boundary of the convex hull of $X$ itself. Let $\Gamma(X)$ be the set of all convex, simple closed curves in $\mathbb{R}^2$...
- 1,677
11
votes
4
answers
958
views
Geometry of the multilagrangian Grassmannian
Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$.
Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...
- 1,783
10
votes
2
answers
4k
views
Morphism between projective varieties
Let $f:X \rightarrow Y$ be a morphism between two smooth projective varieties $X,Y$ which are defined over an algebraically closed field $k$. I am looking for some criteria which guaranties the ...
- 690
9
votes
2
answers
901
views
Subtlety in the definition of the Kobayashi metric
When defining the Kobayashi metric on a connected complex analytic space $X$, one makes the following auxiliary definition:
A holomorphic chain from $x\in X$ to $y\in X$ is a finite sequence of ...
- 1,288
9
votes
2
answers
1k
views
Maximal number of connected components of complement to an affine plane real algebraic curve
Let $X$ be a (singular, reducible) affine plane real algebraic curve of degree $d$.
How we can estimate maximal number of connected components of it's complement in $R^2$ in terms of degree?
- 413
9
votes
2
answers
3k
views
An optimization problem for points on the sphere (master's dissertation)
First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a ...
- 3,612
9
votes
1
answer
529
views
Ricci Curvature on Grassmannian
Suppose $G_r(n)$ is the Grassmannian, which is the collection of all $r$ dimensional subspace in $\mathbb{R}^{n}$ equipped with the usual invariant metric. Let $Ricc(G_r(n))$ be the Ricci curvature ...
- 1,495
9
votes
2
answers
718
views
Generalization of Pascal's theorem to higher dimensions
Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement ...
- 4,484
8
votes
4
answers
4k
views
Proofs for doubly ruled surfaces
Hello,
I am interested in proofs for why the only irreducible doubly ruled surfaces in ${\mathbb R}^3$ are the one sheeted hyperboloid and the hyperbolic paraboloid. While many books and papers state ...
- 1,072
8
votes
2
answers
1k
views
Quadrature of the Lune
What is a good reference for the following result which I believe is proved by Tchebotarev.
There are exactly 5 types of Lunes that are squarable. (Hippocrates produced three and then two more were ...
- 575
8
votes
1
answer
573
views
Do elements of the fundamental group give rise to isometries
Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...
- 103
7
votes
1
answer
347
views
A corollary of the affine Desargues axiom
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
- 41.8k
6
votes
3
answers
539
views
Constructing a field from a spherical building
Tits proved that (sufficiently high rank) spherical buildings arise from an algebraic group and a field, so any building is some $\Delta(G, F)$. He also showed that a building isomorphism $\Delta(G,F)...
- 63
6
votes
2
answers
1k
views
Motivation for Hirzebruch-Jung Modified Euclidean Algorithm
Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in \mathbb{N}$...
6
votes
0
answers
172
views
Does there exist a plane curve such that it has the heart curve as catacaustic?
Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$.
The catacaustic ...
- 565
5
votes
3
answers
572
views
set of centers of sphere inscribed in tetrahedron
Having a sphere and three diffrent point $A,B,C$ on this sphere. Find set of all centers of spheres inscribed in a tetrahedron $ABCD$, where $D$ is some point on the given sphere. The problem reduced ...
- 69
5
votes
4
answers
1k
views
The Icosahedron Equation
$$1728 V^5 + F^3 = E^2 \;.$$
Can anyone point me to a concise, modern derivation and explanation of
the significance of the icosahedron equation, more modern and
concise than Klein's description in ...
- 151k
5
votes
3
answers
548
views
Quadrics containing many points in special position
Suppose $n$ quadric hypersurfaces cut
out $2^n$ distinct points
$p_1,\ldots,p_{2^n}$ in
$\mathbb{P}^n$. What is the maximal
number of points $p_i$ a quadric can
contain without containing ...
- 11.6k
5
votes
3
answers
1k
views
Non-trivial algebraic consequence of an elementary geometric theorem
A well-known theorem in projective geometry states that the three Pascal lines of an arbitrary hexagon inscribed in a quadric intersect in one point. I found an algebraic reformulation, which states ...
5
votes
1
answer
586
views
a general theory of configurations?
Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction ...
- 10.8k
5
votes
0
answers
333
views
Which equation of a Butterfly?
Let $A, B$ be two points and $L$ be a line on the Euclidean Plane. Take two points $J, G$ on the line $L$ such that $JG=constant$. Let $AJ$ meet $BG$ at $P$, $AG$ meet $BJ$ at $Q$, then the locus of ...
- 477
4
votes
2
answers
663
views
Real vs complex surfaces
Hi! My background is in (complex) algebraic geometry. For the first time I'm considering varieties defined over $\mathbb{R}$ and I have some very basic questions about these.
In particular I'm trying ...
- 1,150
4
votes
1
answer
369
views
Comparing two Riemannian metrics on Grassmannian
Let $G_r(n)$ be the real Grassmannian which is the collection of all $r$ dimensional subspace in $\mathbb{R}^n$ equipped with the usual invariant metric $g$.
Let $U_A\in\mathbb{R}^{n\times r}$ and $...
- 1,495
4
votes
1
answer
226
views
A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem
I propose a conjecture variant of Cayley-Bacharach's theorem.
I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know ...
- 1,044
4
votes
2
answers
1k
views
approaches to Apollonius circle problems
I've been looking for solutions to finding the set of circles tangent to two other circles. one circle can be inverted to a line, but two circles can be mapped to a line and a circle
or equivalently ...
- 22.8k
4
votes
1
answer
184
views
What are interesting 3-colorings of the plane without rainbow lines?
This question is about 3-colorings of the plane in which every line is bichromatic (or monochromatic), i.e., there are no three collinear points of different colors. Such colorings trivially exist, ...
- 18.9k
4
votes
0
answers
119
views
Writing the $\ell^{p/(p-1)}$ unit sphere as a semi-algebraic set for $p\in\Bbb N$
The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is
$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$
...
- 13.6k
4
votes
0
answers
239
views
Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$
Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric.
I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
- 1,343
4
votes
0
answers
152
views
Is there an ellipsoid with given outer normals?
Pick two points $(x,0)$ and $(0,y)$ (say $x>0$ and $y>0$). Pick a unit vector $u = (u_1,u_2)$, $v = (v_1, v_2)$, and attach one to each of the points. Provided $u$ and $v$ are "nice" ($v$ needs ...
- 1,069
3
votes
1
answer
270
views
When is a blow-up a non-trivial product?
Suppose $X$ is an algebraic variety and let $Z \subset X$ be a subvariety. Are there some useful criteria under which the blow-up $Bl_Z X$ becomes a nontrivial product $V \times W$ of the algebraic ...
- 690
3
votes
1
answer
363
views
What do convergent sequences of rational functions look like?
Let us consider the projective line over $\mathbb C$ equipped with a nice metric $\eta$ (like the Fubini-Study metric). We can define a metric $\mu$ on rational functions $f: \mathbb P^1 \to \mathbb P^...
- 7,746
3
votes
2
answers
1k
views
Maximum area of intersection between annulus and circle? [closed]
Given two concentric circles $[C_1,C_2 ]$ with radii $(R_1 < R_2) $ creating an annulus; where should a third circle ($C_3$, radius $R_3$) be located such that the area of intersection between ...
- 171