Tight closure/Text exponent/Introduction/Section

The problem with an algorithmic computation of tight closure is that we have to check infinitely many conditions. For a test element ${}z$ (a well established theory) a test exponent is a number ${}e_{0}$ such that ${}zf^{q}\in I^{[q]}$ for all ${}q=p^{e}$ and ${}e\leq e_{0}$ implies ${}f\in I^{*}$. This makes also sense for a restricted class of ideals. But even for parameter ideals nothing substantial is known.

The following variant is more promising and has the same computational effect: Let ${}\tau$ denote the test ideal of ${}R$. We call ${}e_{0}$ a test ideal exponent (for a class of ideals) if

$zf^{q}\in I^{[q]}{\text{ for all }}z\in \tau$

and for all ${}q=p^{e}$ and ${}e\leq e_{0}$ implies ${}f\in I^{*}$. For this one has to know the test ideal, but this is known in many cases. For the class of parameter ideals in the Gorenstein case this works, because then ${}I^{*}=(I:\tau )$ and so we can take even ${}0$ as test ideal exponent.

The methods from above allow us to extend this to homogeneous primary ideals in a standard-graded two-dimensional domain over a finite field. The test ideal exponent is however huge and not suitable for computations. It depends on the genus, the number of ideal generators and most importantly on the number of elements in the field (via the finite number of semistable bundles in the moduli space).

Theorem

Let ${}R$ be a standard-graded, two-dimensional (geometrically) normal Gorenstein domain over a finite field. Fix ${}n$ and suppose that ${}p=4(g-1)(n-1)^{3}$, where ${}g$ is the genus of the curve. Then there exists a test ideal exponent for the class of primary homogeneous ideals generated by ${}n$ elements.

The finite field assumption in the last two statements is necessary. Both proofs rely on the fact that for fixed rank and degree there exists only finitely many semistable sheaves defined over the field with these numerical data.