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Speaker:
Martine Girard and David Kohel, (University of Sydney)
Title:
Weierstrass points and the groups they generate
Abstract:
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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