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Speaker: Martine Girard and David Kohel, (University of Sydney)

Title: Weierstrass points and the groups they generate


Weierstrass points on an algebraic curve are intrinsically defined points which capture aspects of the geometry of the curve. For a hyperelliptic curve -- defined by an equation of the form $y^2 = f(x)$ -- they are fixed points of the involution $i(x,y) = (x,-y)$. For a plane quartic, they are flexes of the curve. For any curve there are just a finite number of these special points. We describe a group generated by the Weierstrass points, in which the relations between points is conveyed by geometric configurations of the points. (As an example, in the case of plane quartics, four points on a line sum to the identity.) For every genus (a measure of the complexity of a curve) we find special curves with configurations of Weierstrass points which determine torsion in the Weierstrass group. We use these special curves to derive the counterintuitive result that a generic curve has no such torsion.

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