Let
be a smooth plane quartic over a discrete valuation field. We characterize the type of
reduction (i.e., smooth plane quartic, hyperelliptic genus 3 curve or bad) over
in terms of the existence of a special plane quartic model and, over
,
in terms of the valuations of certain algebraic invariants of
when the characteristic
of the residue field is not
or
. On the
way, we gather several results of general interest on geometric invariant theory over an
arbitrary ring
in the spirit of work of Seshadri (Advances in Math.26:3 (1977), 225-274). For instance
when
is
a discrete valuation ring, we show the existence of a homogeneous system of parameters
over
.
We exhibit explicit ones for ternary quartic forms under the action of
depending only on
the characteristic
of the residue field. We illustrate our results with the case of Picard curves for which
we give simple criteria for the type of reduction.
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