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Let $X$ be a smooth irreducible projective curve of genus $g$ and gonality 4.We show that the canonical model of $X$ is contained in a uniquely definedsurface, ruled by conics, whose geometry is deeply related to that of $X$. Thissurface allows us to define four invariants of $X$ and hence to stratify themoduli space of four-gonal curves by means of closed irreducible subvarietieswhose dimensions we compute.

Stratification of the moduli space of four--gonal curves