1
vote
1answer
144 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 ...
1
vote
1answer
133 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 ...
15
votes
2answers
275 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 ...
2
votes
0answers
80 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 ...
1
vote
0answers
131 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 ...
0
votes
0answers
77 views

Building an invariant Sn structure from two invariant Zn structures

Take two mathematical structures with a $Z_n$ symmetry (cyclic symmetry). Which are the different ways, in "gluing" these structures, to obtain a mathematical structure with a $S_n$ symmetry ...
4
votes
4answers
704 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 ...
0
votes
0answers
141 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 ...
12
votes
4answers
520 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 ...
5
votes
1answer
263 views

Can a birational morphism between smooth varieties be dominated by smooth blow ups sequences

Suppose $f:X\rightarrow Y$ be an birational morphism between smooth varieties and D is a snc divisor on $Y$. Can we find a smooth blow ups sequence on $Y$ which dominates $f$ such that the preimage of ...
4
votes
1answer
206 views

When is the hull of a space curve composed of developable patches?

Let $C$ be a smooth curve in $\mathbb{R}^3$ that lies entirely on its convex hull, $\cal{H}(C)$. Under what conditions on $C$ is $\cal{H}(C)$ the union of developable surface patches? I believe ...
4
votes
2answers
457 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 ...
3
votes
2answers
494 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 ...
1
vote
1answer
595 views

Covering an arbitrary polygon with minimum number of squares

I have a problem whereby, given an arbitrary polyogn with any number of points, I need to cover the whole area by a number of fixed size squares. I can easily find a set of squares which covers the ...
1
vote
2answers
249 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 ...
1
vote
1answer
326 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 ...
3
votes
0answers
131 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 ...
5
votes
3answers
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 ...
4
votes
1answer
261 views

Blowing up a subvariety - what can happen to the singular locus?

Let $X$ be a variety defined over a number field $k$. If I blow-up along some arbitrary subvariety of $X$, what are the possible outcomes for the dimension of the singular locus of the variety? If ...
6
votes
2answers
527 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?
21
votes
2answers
871 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
vote
0answers
458 views

Maximal disjoint Hyperplanes

Given a set of $n^{r}$ points $X_{r} = \{ x_{1}, \cdots, x_{n^{r}} \}$ occupying a codim $t^{r}$ subspace in $\mathbb{R}^{n^{r}}$. Let $M_{r}$ be the set of $t^{r}$-tuples of these points.. So ...
0
votes
2answers
357 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, ...
3
votes
2answers
388 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 ...
16
votes
4answers
2k views

Classification of PDE

Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for ...
11
votes
2answers
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 ...
5
votes
3answers
490 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 ...
8
votes
2answers
783 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 ...
2
votes
0answers
123 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 ...
4
votes
2answers
789 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 ...
3
votes
1answer
214 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 ...
0
votes
1answer
411 views

How to formulate such problem mathematicaly? (line continuation search)

I have an array of "lines" each defined by 2 points. I am working with only the line segments lying between those points. I need to search lines that could continue one another (relative to some ...
2
votes
0answers
239 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 ...
37
votes
6answers
2k views

What do Weierstrass points look like?

As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you ...
10
votes
4answers
647 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
vote
1answer
237 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 ...
2
votes
3answers
1k views

Various Cartan's Lemmata

I am a bit amazed by "Cartan's Lemma".. I have so far seen it in : Algebraic Geometry sources: Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...
5
votes
1answer
423 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 ...