On the triviality of a family of linear hyperplanes

Parnashree Ghosh (Indian Statistical Institute, Kolkata, India)

14-Oct-2022, 12:00-13:00 (17 months ago)

Abstract: Let k be a field, m a positive integer, $\mathbb{V}$ an affine subvariety of $\mathbb{A}^{m+3}$ defined by a linear relation of the form $x_1^{ r_1}\cdots x_r^{r_m} y = F(x_1,\ldots , x_m, z, t),$ A the coordinate ring of $\mathbb{V}$ and $G = X_1^{ r_1} \cdots X_r^{r_m} Y − F(X_1, \ldots , X_m, Z, T).$ We exhibit several necessary and sufficient conditions for V to be isomorphic $\mathbb{A}^{m+2}$ and G to be a coordinate in $k[X_1, \ldots , X_m, Y, Z, T],$ under a certain hypothesis on F. Our main result immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds. We also describe the isomorphism classes and automorphisms of integral domains of the type A under certain conditions. These results show that for each integer $d\geq 3,$ there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension d in positive characteristic. This is a joint work with Neena Gupta.

Mathematics

Audience: advanced learners


IIT Bombay Virtual Commutative Algebra Seminar

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