Deciding Trigonality of Algebraic Curves (extended Abstract) 1 Classical Results on Trigonality

Let C be an algebraic curve of genus g ≥ 3. Let us assume that C is not hyperelliptic, so that it is isomorphic to its image by the canonical map φ : C → P. Enriques proved in [1] that φ(C) is the intersection of the quadrics that contain it, except when C is trigonal (that is, it has a g 3) or C is isomorphic to a plane quintic (g = 6). The proof was completed by Babbage [2], and later Petri proved [3] that in those two cases the ideal is generated by the quadrics and cubics that contain the canonical curve. In this context, we present an implementation in Magma of a method to decide whether a given algebraic curve is trigonal, and in the affirmative case to compute a map C 3:1 → P whose fibers cut out a g 3 . Our algorithm is part of a larger effort to determine whether a given algebraic curve admits a radical parametrization.