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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
##### Keywords:
moduli space of curves; plane quartic
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##### References:
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