Virtual Commutative Algebra Seminar
Speaker: Parnashree Ghosh, Indian Statistical Institute Kolkata, India
Date/Time: 14 October 2022, 5:30pm IST/ 12:00pm GMT /8:00am ET (joining
time 5:20 pm IST)
Gmeet link: meet.google.com/eap-qswg-xvg [1]
Title: On the triviality of a family of linear hyperplanes
Abstract: Let k be a field, m a positive integer, V an affine
subvariety of $A^{m+3}$ defined by a linear relation of the form $x_1^{
r_1} · · · x_r^{r_m} y = F(x_1, . . . , x_m, z, t),$ A the coordinate
ring of V and $G = X_1^{ r_1} · · · X_r^{r_m} Y - F(X_1, . . . , X_m, Z,
T).$ We exhibit several necessary and sufficient conditions for V to be
isomorphic $A^{m+2}$ and G to be a coordinate in $k[X_1, . . . , X_m, Y,
Z, T],$ 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 type A under certain conditions. These results show that for
each integer d ⩾ 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 joint work with Neena Gupta.
For more information and links to previous seminars,
visit the website of VCAS:
https://sites.google.com/view/virtual-comm-algebra-seminar [2]
Links:
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[1] http://meet.google.com/eap-qswg-xvg
[2] https://sites.google.com/view/virtual-comm-algebra-seminar