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Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields. (English) Zbl 1333.14060
Let \({\mathcal M}_g\) denote the coarse moduli space of (projective, geometrically irreducible, nonsingular algebraic) curves of genus \(g>1\) defined over a field \(k\) of characteristic \(p\). One is often interested to explicitly write down models for a curve corresponding to a point of \({\mathcal M}_g\); this can be done e.g. if a universal family is available which is not the case in general. In the paper under review, new concepts that substitute the notion of universal family are introduced whenever \(p=0\) or \(p>2g+1\); specially the so-called representative family which, given a subvariety \(\mathcal S\) of \({\mathcal M}_g\), is a family of curves \(\mathcal C\to\mathcal S\) whose points are in a natural bijection with those of \(\mathcal S\). It turns out that the existence of a representative family is quite related to the question of whether the field of moduli of a curve is a field of definition.
The authors illustrate their results by working out on families of quartic plane curves \(C\subseteq {\mathbb P}^2\) and by taking \(\mathcal S={\mathcal S}_G\) to be a subvariety of such curves with a given automorphism group \(G\subseteq \mathrm{PGL}_3\). It turns out that the classification of such groups is known if \(p=0\) or \(p\geq 5\); cf. I. V. Dolgachev’s book [Classical algebraic geometry. A modern view. Cambridge: Cambridge University Press (2012; Zbl 1252.14001)]. Then one can compute representative family of \({\mathcal S}_G\) via Galois descent to extensions of function fields. They also give an algorithm to compute the twists of a plane quartic and work out an implementation with MAGMA of some results of the paper for prime fields of order \(p\) with \(7<p<256\).

MSC:
14Q05 Computational aspects of algebraic curves
13A50 Actions of groups on commutative rings; invariant theory
14H10 Families, moduli of curves (algebraic)
14H37 Automorphisms of curves
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