[Paper Review] Hilbert-Kunz multiplicity and reduction mod p

V. Trivedi|ArXiv.org|Jul 29, 2004

Algebraic Geometry and Number Theory4 references4 citations

TL;DR

This paper establishes that the Hilbert-Kunz multiplicity of reductions of an irreducible projective curve in characteristic 0 to positive characteristic $p$ converges to a well-defined limit as $p \to \infty$. Using the Harder-Narasimhan filtration of Frobenius pullbacks of associated vector bundles and asymptotic analysis of normalized slopes, the authors prove the existence of this limit, which is expressed in terms of the slopes of semistable quotients in the strongly semistable HN filtration.

ABSTRACT

We show that the Hilbert-Kunz multiplicities of the reductions to positive characteristics of an irreducible projective curve in characteristic 0 have a well-defined limit as the characteristic tends to infinity.

Motivation & Objective

• To determine whether the Hilbert-Kunz multiplicity of reductions of a projective curve in characteristic 0 converges as the characteristic $p \to \infty$.
• To analyze the behavior of Hilbert-Kunz multiplicities via the Harder-Narasimhan filtration of vector bundles on curves in positive characteristic.
• To establish a well-defined limit for these multiplicities by relating them to normalized slopes of HN quotients in iterated Frobenius pullbacks.
• To show that the limit is computable from invariants of the HN filtration and is independent of the choice of spread.

Proposed method

• The authors construct a finitely generated $\mathbb{Z}$-algebra $A$ and a projective $A$-scheme $X_A$ such that base change to any closed point $s \in \mathrm{Spec}\, A$ yields a model of the original curve in positive characteristic.
• They analyze the Harder-Narasimhan (HN) filtration of the Frobenius pullback $F^{k*}V$ of a vector bundle $V$ on the curve, and define the HN polygon $HNP_{p^k}(V)$ using rank and normalized degree of subbundles.
• They prove that for $p \geq 4(g-1)(\mathrm{rank}\, V)^3$, the vertices of $HNP_{p^{k-1}}(V)$ are retained in $HNP_{p^k}(V)$, ensuring stability of the polygon structure under Frobenius iteration.
• Using a result from Shepherd-Barron, they show that the normalized slope $\underline{\mu}(F_j)/p^k$ of each segment in $F^{k*}V$'s HN filtration satisfies $\underline{\mu}(F_j)/p^k = \underline{\mu}(E_i) + O(1/p)$, where $E_i$ is a quotient from the HN filtration of $V$.
• They compute the area of $HNP_{p^k}(V)$ and prove $\lim_{p \to \infty} \mathrm{Area}\, HNP_{p^k}(V) = \mathrm{Area}\, HNP(V)$, implying convergence of the sum $\sum \widetilde{r}_i \widetilde{a}_i^2$.
• Finally, they define the limit Hilbert-Kunz multiplicity $HKM(R,I)$ as the limit of $HKM(R_s, I_s)$ over closed points $s \in \mathrm{Spec}\, A$, and show it is rational and independent of the choice of spread.

Experimental results

Research questions

• RQ1 Does the Hilbert-Kunz multiplicity of reductions of a projective curve in characteristic 0 converge as the characteristic $p \to \infty$?
• RQ2 Can the limit of Hilbert-Kunz multiplicities be expressed in terms of the Harder-Narasimhan filtration of the associated vector bundle?
• RQ3 How do the normalized slopes of HN quotients in iterated Frobenius pullbacks of a vector bundle behave asymptotically as $p \to \infty$?
• RQ4 Is the limit of Hilbert-Kunz multiplicities independent of the choice of model over $\mathbb{Z}$?
• RQ5 What is the relationship between the Hilbert-Kunz multiplicity in positive characteristic and the limit as $p \to \infty$?

Key findings

• The limit $\lim_{p \to \infty} HK(R_p)$ exists and is equal to $\sum_i \widetilde{r}_i(V_s) \widetilde{a}_i(V_s)^2$, where $\widetilde{r}_i$ and $\widetilde{a}_i$ are the normalized rank and slope of the $i$-th quotient in the strongly semistable HN filtration of $V_s$.
• The limit is rational and computable from the slopes of the HN quotients of the vector bundle on the curve in characteristic 0.
• For large $p$, the normalized slopes $\widetilde{a}_i(V_s)$ converge to the normalized slopes of the HN quotients of the original bundle, with error $O(1/p)$.
• The limit $HKM(R,I)$ is a lower bound for all $HKM(R_s, I_s)$, and equality holds if and only if the HN filtration of $V_s$ is already strongly semistable.
• The convergence is not monotonic: Monsky's example shows $HKM(R_p, I_p)$ can oscillate (e.g., $3$, $3 + 1/(4p^2)$, $3 + 1/(4p^4)$) depending on $p \mod 9$, even as $p \to \infty$.
• In the case of a nonsingular plane curve of degree $d$, the Hilbert-Kunz multiplicity is $\frac{3d}{4} + \frac{l^2}{4d p^{2s}}$, where $s$ is the smallest integer such that $F^{s*}V$ is not semistable, and $l \in \{0, \dots, d(d-3)\}$, $l \equiv pd \pmod{2}$.