20 votes

About the definition of E8, and Rosenfeld's "Geometry of Lie groups"

Here's an easy, direct definition of $E_8$. The compact Lie group $E_8$ is the colimit in the category of topological groups of the following diagram of groups $$ {\scriptstyle\begin{matrix} &SU(...
20 votes
Accepted

Is it a new discovery on conic section?

It suffices to consider the case when $\Omega$ is a circumcircle, so let it be. At first, the points $A_b, A_c, B_c, B_a, C_a, C_b$ lie on a conic if and only if $$ \frac{AB_a\cdot AB_c}{AC_a\cdot ...
  • 90.9k
19 votes
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Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space

This configuration has automorphisms by the symmetric group $S_5$, and can be identified with the planes $a_i = a_j$ ($0 \leq i < j \leq 4$) in the projective 3-space $a_0+a_1+a_2+a_3+a_4 = 0$, by ...
18 votes

About the definition of E8, and Rosenfeld's "Geometry of Lie groups"

The algebraic group $E_8$ is the group of automorphisms of the $E_8$ lattice vertex algebra, by Frenkel-Kac and Segal. This vertex algebra has a self-dual integral form, so the construction works ...
  • 43.4k
18 votes
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Is every smooth projective variety contained in a chain of smooth projective varieties of increasing dimension?

Suppose that $\operatorname{dim}(X)>1$ and that such a chain exists. Since $\operatorname{Pic}(\mathbf{P}^n)\simeq \mathbf{Z}$, the variety $X_{k-1}$ is an ample divisor in $\mathbf{P}^n$, and ...
16 votes

Automorphisms of cartesian products of curves

That is certainly not true. Consider the case that $C$ is an elliptic curve. Then $\text{Aut}(C\times C)$ contains $\text{GL}(2,\mathbb{Z})$ as a subgroup.
16 votes
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When do 27 lines lie on a cubic surface?

It turns out that condition (T) is, indeed, sufficient for the $27$ lines (distinct and intersecting as expected) to lie on a cubic surface. To see this, consider the lines $a_1,a_2,a_3,a_4,a_5$ and $...
  • 23.7k
16 votes
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Projective-invariant differential operator

There's a straightforward abstract answer that you may not like, but, because it clarifies your question and explains a uniform way to answer similar questions, I'll sketch it here. First, consider a ...
15 votes
Accepted

How many subspaces are generated by three or more subspaces in a Hilbert space?

From four subspaces in general position one can generate an infinite number of other subspaces by closing up under joins and meets. This is true even for subspaces of $\mathbb{R}^3$ (any field of ...
  • 51k
15 votes

Which weighted projective spaces (and their finite quotients) are local complete intersections?

Regarding your question about weighted projective spaces, a lot is known about them, see for instance [1] and [2]. In particular, any weighted projective space $\mathbb{P}(\mathcal Q)$ is irreducible,...
15 votes
Accepted

Geodesic preserving diffeomorphisms of constant curvature spaces

For $\mathbb{R}^n$: the fundamental theorem of projective geometry (proof: https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $\mathbb{R}^n$ taking lines to lines are ...
  • 23.7k
14 votes
Accepted

Transitive actions of finite subgroups of ${\rm GL}(n,\Bbb Z)$ on projective geometries

Probably final revision: I am indebted to Dave Witte-Morris, who added a reference to a refinement of Zsigmondy's Theorem by W. Feit, of which I was unaware, and pointed out that consequently, a ...
14 votes
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What are Sylvester-Gallai configurations in the complex projective plane?

Yes, there are other Sylvester-Gallai configurations in $\mathbb{P}^2(\mathbb{C})$. Apart from the Hesse configuration (that contains $9$ points) the minimum number of points for a non-collinear ...
13 votes
Accepted

Equivariant Almost Complex Structures on the Full Flag Manifolds

Actually, though this may seem pedantic, there are two almost-complex structures on $\mathbb{CP}^m$ that are invariant under $\mathrm{SU}(m{+}1)$, namely the 'standard' one and its conjugate. Of ...
13 votes

About the definition of E8, and Rosenfeld's "Geometry of Lie groups"

Recently Lusztig gave a much simpler definition of $E_8$ (and all the simple Lie algebras/groups) that avoids the usual sign issue with the standard Chevalley/Serre construction. See Lusztig - On ...
  • 19.8k
13 votes

Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space

With the aid of the answer I now managed to find a better realization. At the expense of some of the symmetries it can be nicely drawn in 3d space - it is just the barycentric subdivision of a ...
13 votes
Accepted

Presentations of $\mathbf{PGL}_3(\mathbb{F}_2)$ by three involutions

This cannot be done. Let $G_1$ and $G_2$ be the groups $$G_1 = \langle a,b,c | a^2 = b^2 = c^2 = (ab)^4 = (bc)^4 = (ac)^2 \rangle$$ $$G_2 = \langle a,b,c | a^2 = b^2 = c^2 = (ab)^3 = (bc)^3 = (ac)^3 \...
13 votes
Accepted

Planes in Lagrangian Grassmannians

This is, indeed, true. To prove this, assume we have an embedding $\mathbb{P}^2 \to \operatorname{LGr}(V)$ (where $V$ is a symplectic vector space). Let $U \subset V \otimes \mathcal{O}$ be the ...
  • 32.8k
13 votes
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The map $k \mapsto \mathbf{PGL}_2(k)$

What's written above is true for all $n, n_1 \geq 2$ and all skew fields except two finite cases: $PSL(2, \Bbb F_7) \cong PSL(3, \Bbb F_2)$, $PSL(2, \Bbb F_4) \cong PSL(2, \Bbb F_5)$. (V. M. Petechuk, ...
  • 3,703
12 votes
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Natural examples of Reverse Mathematics outside classical analysis?

Reverse math usually means work in subsystems of arithmetic. That goes a bit beyond analysis---Simpson's book has plenty of classic results from the theory of countable groups, rings, and fields, and ...
12 votes
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Looking for reference or proof to some facts stated on Anand Pillay's book

In the notes on chapters at the end of Geometric Stability Theory, I give references. For Fact 1.11 it is Doyen and Hubaut, Finite regular locally projective geometries, Math. Zeitschrift, 1971. ...
12 votes

Can the projective line be provided with a ring structure?

Here is a less algebraic and more topological answer: it's known that any compact (Hausdorff) topological ring must be totally disconnected. In particular, there's no hope for either $\mathbb{RP}^1$ ...
12 votes
Accepted

Degree of secant varieties of Veronese varieties

The secant variety $Sec_k(V^n_2)$ is the variety parametrizing $(n+1)\times (n+1)$ symmetric matrices modulo scalar of rank at most $k$ that is of corank at least $n+1-k$. Then by Proposition 12(b) in ...
  • 6,732
12 votes
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What is the automorphism group of the projective line minus $n$ points?

For $n \geq 5$, we can describe the locus of configurations that have nontrivial automorphisms. To do this, note that if there is any nontrivial automorphism, there is an automorphism of order $p$ ...
  • 122k
12 votes
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A question on a Macaulay2 computation

Commutative algebra is NOT the same as algebraic geometry, especially projective algebraic geometry. The variety in $\mathbb{P}^9$ defined by $I$ and the variety in $\mathbb{P}^9$ defined by $I_0$ are ...
12 votes
Accepted

p-adic analogue of octonions

Defining and classifying the octonion algebras (composition algebras of dimension $8$) over fields $k$, or, in more sophisticated terms, computing the Galois cohomology set $H^1(k, G_2)$, is the topic ...
  • 23.7k
11 votes

Who first proved the fundamental theorem of projective geometry?

The version you state is definitely a 20th century development, only marginally related to Von Staudt's theorem. Here is a translation of the relevant section of Karzel & Kroll's Geschichte der ...
11 votes
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Generalization of the rigidity lemma in birational geometry

EDIT: I've just realized that this holds under somewhat weaker assumptions. It is not necessary that the fibers of $g$ are connected. EDIT#2: Apparently, in my previous edit I weakened the conditions ...
11 votes
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Fundamental groups of complements of divisors in $\mathbb P^2$

I'd leave this as a comment, but I don't have enough reputation. Consider the long exact sequence in homology of the pair $(\mathbb{P}^2, \mathbb{P}^2-D)$. Since $H_1(\mathbb{P}^2,\mathbb{Z}) = 0$ and ...
  • 666
11 votes
Accepted

Non-isomorphic projective planes on $\omega$

You ask for the number of isomorphism classes of projective planes on $\omega$. I claim that it is exactly $2^{\aleph_0}$. It is at most $2^{\aleph_0}$. Indeed, a projective plane on $\omega$ can ...
  • 23.7k

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