Seminar Algebra and Geometry: Adrien Dubouloz (Université de Bourgogne)

On the Cancellation Problem for algebraic tori

A much less studied version of Zariski Cancellation Problem asks whether two algebraic varieties which become isomorphic after taking their products with an affine algebraic torus are isomorphic themselves. It is easy to see that the answer is positive when the two varieties are smooth algebraic curves. For higher dimensional varieties, the problem is more subtle, intimately related to the geometry of families of affine rational curves on them. The only available general result so far is that "cancellation" holds provided that one the two varieties is not dominantly covered by images of the punctured affine line, a property satisfied for instance by varieties of log-general type. After a brief survey of various aspects of the problem, I will present some strategies to construct smooth affine factorial counter-examples to cancellation in any dimension bigger or equal to two.

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