Maximally symmetric smooth projective varieties in CP^2 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:34:20Z http://mathoverflow.net/feeds/question/19661 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19661/maximally-symmetric-smooth-projective-varieties-in-cp2 Maximally symmetric smooth projective varieties in CP^2 Daniel Asimov 2010-03-28T20:12:00Z 2010-03-28T22:14:56Z <p>Let P(X,Y,Z) be a homogeneous polynomial in &#8450;[X,Y,Z] whose locus M in &#8450;&#8473;<sup>2</sup> is a nonsingular curve of genus &ge; 2.</p> <p>Define M to be <em>maximally symmetric</em> if the following is <strong>not</strong> true: </p> <hr> <p>There exists a continuous family {&nbsp;P<sub>t</sub> &nbsp; |&nbsp; t &#8714; [0,1]&nbsp;} of homogeneous polynomials in &#8450;[X,Y,Z] such that 1), 2), and 3) hold:</p> <p>1) P<sub>0</sub> = P.</p> <p>2) The locus M<sub>t</sub> in &#8450;&#8473;<sup>2</sup> of each P<sub>t</sub> is nonsingular.</p> <p>3) There is a group G such that the ambient isometry groups <br/>G<sub>t</sub> := Isom<sub>A</sub>(M<sub>t</sub>) are all isomorphic to G for 0 &le; t &lt; 1, but G<sub>1</sub> contains G as a proper subgroup.</p> <p>Here the "ambient isometry group" Isom<sub>A</sub>(M<sub>t</sub>) of a projective curve M in &#8450;&#8473;<sup>2</sup> means the subgroup of Isom(&#8450;&#8473;<sup>2</sup>) = PSU(3) that carries M to itself.</p> <hr> <p>Question: I'd like pointers to the literature regarding what may be known about a classification of such "maximally symmetric" projective curves up to ambient isometry, their defining polynomials, and <em>especially</em> their ambient isometry groups.</p> <hr> http://mathoverflow.net/questions/19661/maximally-symmetric-smooth-projective-varieties-in-cp2/19672#19672 Answer by Bjorn Poonen for Maximally symmetric smooth projective varieties in CP^2 Bjorn Poonen 2010-03-28T22:10:28Z 2010-03-28T22:10:28Z <p>By the same averaging trick that shows that finite-dimensional complex representations of a finite group are unitary with respect to some inner product, your question is equivalent to the one obtained by replacing ambient isotropy groups with linear automorphism groups in the sense of algebraic geometry. Here <em>linear</em> means induced by a linear automorphism of $\mathbf{P}^2$. Actually, for a smooth plane curve of degree $d>3$, all automorphisms are linear, by</p> <p>H. C. Chang, On plane algebraic curves, <em>Chinese J. Math.</em> <strong>6</strong> (1978), 185-189.</p> <p>Fix $d>3$. Let $\mathcal{H}_d$ be the moduli space of smooth degree-$d$ curves in $\mathbf{P}^2$, so $\mathcal{H}_d$ is an open subscheme of some projective space. Then there is a stratification of $\mathcal{H}_d$ into finitely many locally closed subschemes such that the automorphism group is constant on each piece. (This could also be stated in terms of the automorphism group scheme of the universal curve over $\mathcal{H}_d$.) In these terms, you are asking for the $0$-dimensional strata, or equivalently the smooth plane curves such that in a punctured neighborhood of the corresponding point of $\mathcal{H}_d$ the automorphism group is strictly smaller.</p> <p>The analogous question with $\mathcal{H}_d$ replaced by the full moduli space $\mathcal{M}_g$ of curves of genus <code>$g&gt;1$</code> has been much studied. The direct analogue of your maximally symmetric curves in this setting are the curves said to have "many automorphisms" in Section 3 of the article</p> <p>Jürgen Wolfart, The obvious part of Belyi's theorem and curves with many automorphisms, pp. 97-112 in: <em>Geometric Galois actions 1</em>, edited by L. Schneps and P. Lochak, LMS Lecture Notes Series <strong>342</strong>, Cambridge Univ. Press, 1997.</p> <p>Wolfart's article contains many references to related work, and mentions some nice theorems. For instance: a smooth projective curve of genus greater than $1$ over $\mathbf{C}$ has many automorphisms if and only if it is a Galois cover of $\mathbf{P}^1$ ramified only above $0,1,\infty$.</p>