All Questions
Tagged with ag.algebraic-geometry mg.metric-geometry
86 questions
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 ...
- 6,210
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 $\pi_1(X) \subset Aut(\tilde X)$.
...
- 2,727
2
votes
1
answer
414
views
Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain
Let $H$ be a bounded symmetric domain.
What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?
- 26.3k
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 ...
- 7,902
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, ...
- 12.1k
1
vote
1
answer
176
views
Helly's number from biconvex functions
Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq \...
- 32.4k
2
votes
1
answer
171
views
Helly's Theorem for Biconvex Sets
Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
- 148k
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 ...
- 386
2
votes
0
answers
83
views
Largest subsets of quadrics consisting of "nonorthogonal" vectors
Assume we have an $A$-module $M$, and a quadratic form $q : M \to A$. Recall that it means that
1) $q (a m) = a^2 q (m)$ for all $a \in A$ and $m \in M$, and
2) $B_q (x, y) := q (x + y) - q (x) - q ...
- 311
1
vote
0
answers
142
views
Relationship between stabilizers of a general point and a boundary point
Let $V$ be an n-dimensional complex vector space, and $u\in S^nV$ be a polynomial, $G(u)$ be the stabilizer of $u$ in $GL(V)$. Let $[v]\in\overline{GL(V)\cdot[u]}\subset\mathbb{P}(S^nV)$, but $v\notin ...
- 105
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 ...
- 68.9k
1
vote
0
answers
167
views
How do Hodge classes for Calabi-Yau 4-folds compare with the classes for tori?
Let $X$ be a Calabi-Yau 4-fold, i.e., a connected 4-dimensional compact Kahler manifold with $K_{X} \cong \mathscr{O}_{X}$ and $h^{i} (X,O_{X} )= 0$ for $0 \lt i \lt 4$.
Given a general 4-...
- 45.7k
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 ...
CommunityBot
- 1
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 ...
- 21.3k
1
vote
2
answers
339
views
Smooth a matrix
I have a matrix in which each element contains the coordinates of a 3D surface. Sometimes, some points will be "out of line" meaning that they will not conform to the general shape. For example you ...
- 14.4k
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 ...
CommunityBot
- 1
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 ...
- 41
1
vote
1
answer
698
views
Number of connected components of complement to a reducible real algebraic hypersurface.[EDITED]
Let $X_1,\ldots X_k$ be irreducible(may be singular) affine real algebraic hypersurfaces in $R^n$ with $x_1,\ldots, x_k$ connected components, respectively.
Let $G_1,\ldots, G_l$ be their ...
CommunityBot
- 1
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
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
0
votes
2
answers
376
views
Mean Three Dimensional Shape of Surfaces
If I have $n, 1 < i < n, $ surfaces composed of $f_i$ faces and $v_i$ vertices, how would I go about finding the average surface?
(I'm unsure what I mean by average - intuitively it's obvious, ...
CommunityBot
- 1
1
vote
0
answers
2k
views
Fitting an ellipse to an arbitrary polygon
Hello,
I'd like an algorithm for fitting an ellipse to a polygon. This polygon may be convex or concave. I've read about fitting an ellipse inside or outside a polygon (maximal and minimal of size, ...
- 11
3
votes
2
answers
733
views
Simultaneous resolutions and deformations of simple singularities
Let $X\to \Delta$ be a flat family of complex surfaces with at most a finite number of singularities of simple type, where $\Delta$ is a complex domain in $\mathbb C$.
Here simple type means ...
- 438
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 ...
- 28.9k
1
vote
0
answers
146
views
Are spherical codes algebraic?
Jeffrey Wang in Section 4.2 writes "Since a code is the solution to a number of polynomial equalities between the shortest edges, the coordinates of each rigid point in the code are algebraic and lie ...
- 11
2
votes
0
answers
160
views
Tubular neighborhood growth of zero set of polynomial of bounded degree in the torus
This question is related to my related post:
Volume growth of tubular neigbhorhood of critical values of an algebraic/differentiable map
The setting here is as follows:
Let $p: \mathbb{R}^{2k} \to \...
CommunityBot
- 1
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
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 ...
- 5,137
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 ...
- 22.3k
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
2
votes
0
answers
254
views
Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
CommunityBot
- 1
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 \...
- 28.9k
1
vote
1
answer
335
views
Systems of conics
It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...
- 1,618