Let $K$ be an algebraically closed field of characteristic $p>0$ and let $I\subset K[x_{1},\dots,x_{n}]$ be an ideal generated by (homogeneous) polynomials of degree $d$. Assume that $I$ is reduced, that is, its nilpotency index (aka the Noether exponent; the smallest integer $e$ such that $(\sqrt{I})^{e}\subseteq I$) is $1$. Let $f_{1},\dots, f_{k}\in K[x_{1},\dots,x_{n}]$ be linear homogeneous polynomials.
By results of Kollár (Sharp effective Nullstellensatz), and later improvements by Jelonek (On the effective Nullstellensatz) it is possible to bound the nilpotency index of $(I,f_{1},\dots,f_{k})$ by $d^{n}$. I am wondering if the knowledge that the nilpotency index of $I$ is $1$ can be used to improve this without knowing anything else about $I$:
Is there a bound on the nilpotency index of the ideal $(I,f_{1},\dots,f_{k})$ which is better than $d^n$? Does it help if we know that the ideal defines an affine group scheme, that is, if $K[x_{1},\dots,x_{n}]/(I,f_{1},\dots,f_{k})$ is a Hopf algebra?
Even if we only have a bound similar to $d^{n}$ in general, I am interested in bounding the possible prime divisors of the nilpotency index of $(I,f_{1},\dots,f_{k})$. We certainly need at least the prime divisors of $d$ in general, but are these enough?
