Timeline for Deformation of isolated singularities and non zero divisors

7 events

when toggle format	what		by	license	comment
Aug 30, 2022 at 17:26	vote	accept	Serge the Toaster
Apr 28, 2022 at 13:07	answer	added	Jason Starr		timeline score: 4
Apr 28, 2022 at 10:33	comment	added	Jason Starr		In that case, what about $F=3(x_1^5+x_2^5)-5t^2x_1^3x_2^3$? Modulo $J=\langle \partial_1(F),\partial_2(F) \rangle$, we have $x_1^4\equiv t^2x_1^2x_2^3$, so that $x_1^8\equiv t^4x_1^4x_2^6$. We also have $x_2^4 \equiv t^2 x_1^3x_2^2$, so that $x_2^6\equiv t^2x_1^3x_2^4 \equiv t^4x_1^6x_2^2$. Thus, $x_1^8\equiv t^4 x_1^4 x_2^6 \equiv t^8 x_1^{10} x_2^2$, i.e., $x_1^8(1-t^8x_1^2x_2^2) \equiv 0$. Thus, $x_1^8\equiv 0$. Similarly, $x_2^8\equiv 0$. So $\partial_t(F) = -10tx_1^3x_2^3$ is nilpotent modulo $J$. Yet $x_1^3x_2^3$ is not zero modulo $I$. So $t$ is a zero divisor modulo $I$.
Apr 28, 2022 at 6:08	history	edited	Serge the Toaster	CC BY-SA 4.0	deleted 4 characters in body
Apr 28, 2022 at 6:08	comment	added	Serge the Toaster		@Z.M $\mathbb{C}\{x_1, \dots, x_n\}$ is the ring of germs (at the origin) of analytic maps in $n-$variables. $F$ is a deformation of $f$ if $F\_{t=0}=f$ (as you said).
Apr 27, 2022 at 16:20	comment	added	Z. M		What does the notation $\mathbb C\{x_1,\ldots,x_n\}$ mean? Formal power series which has a positive convergence radius? And what is "a deformation"? Does that mean that $F\vert_{t=0}=f$?
Apr 27, 2022 at 13:45	history	asked	Serge the Toaster	CC BY-SA 4.0