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(...
André Henriques's user avatar
20 votes
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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 ...
Fedor Petrov's user avatar
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 ...
Noam D. Elkies's user avatar
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 ...
S. Carnahan's user avatar
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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 ...
Piotr Achinger's user avatar
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 $...
Gro-Tsen's user avatar
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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 ...
Robert Bryant's user avatar
16 votes
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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 ...
Ben McKay's user avatar
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16 votes
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A projective plane in the Euclidean plane

Let $\ P^2(\mathbb Q)\ $ and $\ P^2(\mathbb R)\ $ be the projective planes over rationals and reals. Let $\,\ L\subseteq P^2(\mathbb R)\,\ $ be a straight line in the real plane such that $$ L\cap P^2(...
Wlod AA's user avatar
  • 4,676
14 votes
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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 ...
Geoff Robinson's user avatar
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 ...
Francesco Polizzi's user avatar
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 ...
Sam Hopkins's user avatar
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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
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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 \...
David E Speyer's user avatar
13 votes
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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 ...
Sasha's user avatar
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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, ...
Denis T's user avatar
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12 votes
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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 ...
Puzzled's user avatar
  • 8,832
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 $p$ ...
Will Sawin's user avatar
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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 ...
Alexander Woo's user avatar
12 votes
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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 ...
Gro-Tsen's user avatar
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12 votes
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Is there a non-singular cubic surface that has a point where four lines intersect?

No, this is not possible. If p is a smooth point on any surface S, and is contained in a line l on S, then l is contained in the tangent plane at p, call it T_p. Now if S is a cubic then it ...
Noam D. Elkies's user avatar
12 votes
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Fermat cubic hypersurfaces over finite fields

A. Weil, in "Numbers of solutions of equations in finite fields" (Bull. Am. Math. Soc. 55, 497-508 (1949)), proved that the number of $\mathbb{F}_q$-points when $q\equiv 1 \bmod 3$ is $$\...
Ofir Gorodetsky's user avatar
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 ...
Sándor Kovács's user avatar
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 ...
K.K.'s user avatar
  • 666
11 votes
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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 ...
Gro-Tsen's user avatar
  • 29.7k
11 votes
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Largest number of points one can pick in finite projective space without getting three on a line

The term for such sets is "caps". The problem you ask was posed by Bose ("Mathematical theory of the symmetrical factorial design", Sankhyā 8 (1947) 107–166), and is important in relation to coding ...
Gro-Tsen's user avatar
  • 29.7k
11 votes
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Intersections in $\mathbb{P}^1\times\mathbb{P}^1$

The curve $V_i$ is given by the vanishing of a polynomial $F_i(x_1,x_2,y_1,y_2)$ that is homogeneous in $x_1,x_2$ of degree $d_{i,1}$ and homogeneous in $y_1,y_2$ of degree $d_{i,2}$. Then counting ...
Joe Silverman's user avatar
11 votes
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Singular cardinal $\kappa$ with projective plane such that all edges have cardinality $<\kappa$

The answer is no. Let $\kappa$ be any infinite cardinal, regular or singular, and assume for a contradiction that there is a set $E\subseteq\mathcal P(\kappa)$ satisfying your conditions. I will call ...
bof's user avatar
  • 11.1k
11 votes

A geometric definition of the addition law on abelian surfaces

This must be standard, I don't have a reference but the construction is easy: let $y^2=f(x)$ be a genus 2 hyperelliptic curve with $f$ squarefree of degree $5$ or $6$. As a set the Jacobian is the ...
Henri Cohen's user avatar
  • 11.3k

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