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8h
comment Separating the points of projective spaces with real-analytic functions
@RobertBryant: You are completely right, thank you for the remark. Since I was thinking to the case of two points (in which general position automatically holds), I wrote an imprecise statement for the general case.
8h
comment Separating the points of projective spaces with real-analytic functions
@Alex M: This is just elementary projective geometry. Any subset of $n+2$ points in $\mathbb{P}^n(\mathbb{K})$ can be sent to any other by using a projectivity. In other words, the group of projective automorphisms is $(n+2)$-transitive. Of course a projectivity is real-analytic, since it is induced by a linear map of $\mathbb{K}^{n+1}$.
8h
comment Separating the points of projective spaces with real-analytic functions
The automorphism group of the projective space is $2$-transitive. Then, since you can separate a pair of points with real analytic functions, you can separate all pairs: just apply a suitable automorphism sending the second pair to the first one.
Feb
4
revised How can we detect the existence of almost-complex structures?
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Feb
1
accepted Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
Feb
1
comment Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
@abx: I checked the computations and everything seems ok, thank you again for your help. Just to be pedantic: if $Q$ is the invariant quadric, one obtains an invariant cubic of the form $V_3$ by taking a general element in the invariant subspace of $\mathsf{S}^3 H^0(C_4, \, K)$ and adding suitable multiples of $x_0Q$ and $x_1Q$ in order to get rid of the monomials $x_0x_1x_3$, $x_0x_2x_3$.
Feb
1
comment Braids with an infinite number of strings
References here: arxiv.org/pdf/math/0201303v3.pdf
Jan
30
comment the term “minimal degree” in a set
Think about the usual division of polynomials that we learn in school. The process go on, until the degree of the remainder is strictly less than the degree of the divisor (or the remainder is 0). That said, this is not a research question so I vote to close.
Jan
29
awarded  Enlightened
Jan
29
awarded  Nice Answer
Jan
29
revised Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
edited title
Jan
29
comment Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
@Holonomia: well, I see that the existence of an invariant cubic in the ideal of $C_4$ easily follows using David Speyer's averaging argument.
Jan
29
comment Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
@Holonomia: the quadric through $C_4$ is unique (so necessarily invariant), but the cubic is not: there is a $5$-dimensional vector space of them.
Jan
29
comment Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
(2) How can I detect the space of invariant cubics through the canonical curve? Using Koszul resolution I can compute $$h^0(\mathscr{I}_{C_4})=5,$$ and it seems to me that there are $8$ invariant monomials, namely $$x_0^3, \, x_0^2x_1, \, x_0x_1^2, \, x_1^3, \, x_0x_2x_3, \, x_1x_2x_3, x_2^3, \, x_3^3.$$ Why you say Writing down the invariant quadric and cubic(s) containing $C_4$ leads to your formulas? How can I check that there exist an invariant cubic containing $C_4$ and whose equation does not contain the invariant monomials $x_0x_2x_3, \, x_1x_2x_3$?
Jan
29
comment Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
Thank you for the answer, it is really helpful. I have a couple of questions. (1) Your computations show that the curve $C_4$ embeds in $\mathbb P^4$ and must be invariant by the action $$[x_0: x_1 : x_2 : x_3] \mapsto [x_0: x_1 : \xi x_2 : \xi^2 x_3].$$ On the other hand, it is a canonical curve, so it is intersection of a (unique) quadric and a cubic. But why both the quadric and the cubic must be invariant, too?
Jan
29
revised Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
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Jan
29
asked Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action
Jan
26
revised Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?
deleted 68 characters in body; edited title
Jan
26
answered Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?
Jan
25
revised Does anyone know the classification of fourth order surfaces?
added 11 characters in body