What is the automorphisms group of the weighted projective space $\mathbb{P}(a_{0},...,a_{n})$ ? Consider the simplest case of a weighted projective plane, take for instance $\mathbb{P}(2,3,4)$; any automorphism has to fix the two singular points. Consider a smooth point $p\in\mathbb{P}(2,3,4)$. What is the subgroup of the automorphisms of $\mathbb{P}(2,3,4)$ fixing $p$ ?
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$\begingroup$ Will anyone answer in general what an automorphism group for a weighted projective space looks like? $\endgroup$– Stanley Yao XiaoCommented Sep 23, 2012 at 20:48
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$\begingroup$ It is a toric variety, so you could try Cox' paper on the homogenous coordinate ring of a toric variety.. $\endgroup$– J.C. OttemCommented Sep 24, 2012 at 16:19
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3 Answers
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The automorphism group is the quotient of the automorphism group of the corresponding graded algebra by 1-dimensional torus acting by rescaling. In the particular case of $P(2,3,4)$ the graded algebra is $A = k[x_2,x_3,x_4]$ with $\deg x_i = i$. Note that any automorphism should take $x_2 \to a x_2$, $x_3 \to b x_3$ (since those are only elements of $A$ of degree 2 and 3) and $x_4 \mapsto c x_4 + d x_2^2$. So, the group can be written as $((k^*)^3 \ltimes k) / k^*$, where $\ltimes$ stands for the semidirect product.
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2$\begingroup$ I am a bit confused about this. Consider $\mathbb P(1,2)$. This should be the same as $\mathbb P^1$. So the automorphism group is $\mathbb PGL(2)$. But with the method you describe, wouldn't you get a different answer? $\endgroup$– mrwCommented Jun 12, 2011 at 19:08
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4$\begingroup$ I consider the weighted projective space as the quotient stack of a vector space (minus zero) by the torus action. This is not the same as a quotient in the category of schemes, and so the automorphism group is different. In your example, the quotient stack is $P^1$ but with an orbifold $Z/2Z$ point. The automorphism group should preserve the point. If you compare the subgroup of $PGL(2)$ preserving a point to the group analogous to the one in my answer you will see they are the same. $\endgroup$– SashaCommented Jun 12, 2011 at 19:53
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$\begingroup$ ok. Thanks for the explanation. I also wanted to mention that the OP asked a slight different. He is asking what are the automorphisms that fix a smooth fixed point p. $\endgroup$– mrwCommented Jun 13, 2011 at 13:52
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$\begingroup$ The weighted projective plane $\mathbb{P}(2,3,4)$ is ismorphic to $\mathbb{P}(1,2,3)$. Following your method I find different automorphisms for $\mathbb{P}(1,2,3)$ and $\mathbb{P}(2,3,4)$. Where am I wrong ? $\endgroup$– PuzzledCommented Jul 7, 2011 at 15:03
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The group Aut(WPS) is studied in "Classes d'idéaux et Groupe de Picard des Fibrés projectifs tordus " §8.2.,8.3.
K-Theory 2 (1989),559-578 (A. AL-AMRANI)
A. Al-Amrani, May 16, 2013.
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2$\begingroup$ Here is a short description of Aut(WPS), in English ! Aut(WPS) is completely computed in the article above.If the WPS is Proj(S), where S is the corresponding (weighted) A-graded algebra (here A is assumed a field), then consider G = Aut. of S as grad.-alg.. An explicit description of G is given, with an A* (units of A) weighted operation on. It is shown that the natural morphism from G to Aut(WPS) induces an isomorphism between G/A* and Aut(WPS). Remark : there is a weighted projective linear description of G/A* ( a generalization of PGL(n,A)) . A.Al-Amrani. $\endgroup$ Commented May 17, 2013 at 7:15
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$\begingroup$ About last Sasha'comment above, P(2,3,4) and P(1,2,3) are isomorphic by reduction of the weights ( proved first by Charles Delorme in "Espaces projectifs anisotropes" ). The computation of Aut(PWS) I mentioned myself is done after reduction of the weights ( which is shown to be the best possible ). $\endgroup$ Commented May 22, 2013 at 14:22
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$\begingroup$ In the paper, did you prove the result regarding $Aut(WPS)$ only for a WPS over a field? It seems otherwise the isomorphism $f^* \mathcal{O}_{WP}(k) \to \mathcal{O}_{WP}(k)$ wouldn't be quite correct. Namely, I think it would have to be $f^* \mathcal{O}_{WP}(k) \to \mathcal{O}_{WP}(k) \otimes \mathcal{L}$ where $\mathcal{L}$ is the pullback of a line bundle on the base. $\endgroup$– userCommented Aug 11, 2019 at 18:42
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$\begingroup$ @user7090 :Yes, in the paper as published above, Aut (WPS) is studied over a field A. The original preprint (Strasbourg IRMA June 1983) contains a paragraph more (§8.4) giving a marticial interpretation of G/A*. It ends with an ?isomorphism? : W(X) --->Aut(WPS) over a scheme X . It is mentioned that for X=Spec(A), A factoriel ring, the morphism is an isomorphism, since Cl(WPS over A) = Z, is generated by O(1) over WPS.In the same way as over a field. If you are interested,I can send you a copy of my hole preprint. $\endgroup$ Commented Aug 15, 2019 at 16:00
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Just to complement Al-Amrani's answer (and to self-promote a little bit): the automorphism 2-group of the weighted weighted projective stack $\mathbb{P}_S(a_1,\ldots,a_n)$ over an arbitrary base scheme $S$ has been computed, and its structure elaborated, in https://doi.org/10.1016/j.jalgebra.2017.05.002
If I am not misunderstanding something, Al-Amrani's result can be interpreted as saying that the $\pi_0$ of the 2-group of automorphisms of the stack is isomorphic to the group of automorphisms of its coarse moduli space (namely, the corresponding weighted projective space).
