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• 90.9k
Accepted

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 ...

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
Accepted

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 ...
• 14.4k
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.