Poncelet-type theorems and points of finite order on a curve in its Jacobian
Thompson, Benjamin L.
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For nearly three centuries mathematicians have been interested in polygons which simultaneously circumscribe and inscribe quadrics. They have shown in many contexts (real, complex, non-euclidean, higher dimensional, etc.) that such polygons may be ``rotated'' while maintaining their circum-inscribed quality. Of particular interest has been conditions on the quadrics which guarantee the existence of such polygons. In 1854 Arthur Cayley provided conditions for closure general to polygons of any size in the complex projective plane. We show that under suitable circumstances the curve, defined by Cayley's conditions, on a fibration of Jacobians over the space of families of quadrics is a reducible curve, particularly in genus two. We may infer additional information about points of finite order on the Jacobians based on the component of the reducible curve in which they lie. Using this information we are able to accomplish two tasks. First we provide sufficient closure conditions for Poncelet's Great Theorem in which each vertex of the polygon lies on a distinct quadric. Next, for a polygon circum-inscribed in quadrics in ℙ^3, we provide additional sufficient conditions for closure beyond what mathematicians had previously believed to be necessary and sufficient.
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