BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Neena Gupta (Indian Statistical Institute\, Kolkata) DTSTART:20200731T120000Z DTEND:20200731T130000Z DTSTAMP:20260423T021001Z UID:VCAS/14 DESCRIPTION:Title: On the triviality of the affine threefold $x^my = F(x\, z\, t)$ - Part 2 \nby Neena Gupta (Indian Statistical Institute\, Kolkata) as part of IIT B ombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nIn this talk we w ill discuss a theory for affine threefolds of the form $x^my = F(x\, z\, t )$ which will yield several necessary and sufficient conditions for the co ordinate ring of such a threefold to be a polynomial ring. For instance\, we will see that this problem of four variables reduces to the equivalent but simpler two-variable question as to whether F(0\, z\, t) defines an e mbedded line in the affine plane. As one immediate consequence\, one read ily sees the non-triviality of the famous Russell-Koras threefold $x^2y+x +z^2+t^3=0$ (which was an exciting open problem till the mid 1990s) from t he obvious fact that $z^2+t^3$ is not a coordinate. The theory on the abov e threefolds connects several central problems on Affine Algebraic Geometr y. It links the study of these threefolds with the famous Abhyankar-Moh “Epimorphism Theorem” in characteristic zero and the Segre-Nagata line s in positive characteristic. We will also see a simplified proof of the triviality of most of the Asanuma threefolds (to be defined in the talk) a nd an affirmative solution to a special case of the Abhyankar-Sathaye Conj ecture. Using the theory\, we will also give a recipe for constructing inf initely many counterexample to the Zariski Cancellation Problem (ZCP) in p ositive characteristic. This will give a simplified proof of the speaker's earlier result on the negative solution for the ZCP.\n LOCATION:https://researchseminars.org/talk/VCAS/14/ END:VEVENT END:VCALENDAR