HILBERT-KUNZ MULTIPLICITIES AND VECTOR BUNDLES OVER
CURVES
V. TRIVEDI

1. abstract
Let (R, m) be a Noetherian local ring (respectively a standard graded ring over a field kwith the graded maximal ideal m) of dimension d and of prime characteristic p > 0, and let I
be an m-primary ideal (respectively a homogeneous ideal of R, of finite colength). Then one
defines the Hilbert-Kunz function of R with respect to I asnHKR,I (pn ) = `(R/I (p ) ),wherenI (p ) = n-th Frobenius power of I= ideal generated by pn -th powers of elements of I.
The associated Hilbert-Kunz multiplicity (HK multiplicity for the sake of abberviation) is defined
to beHKR,I (pn ).
eHK (R, I) = limn→∞pnd
In the graded situation we denote alternatively HK multiplicity eHK (R, I) by eHK (X, L, I),
where X = Proj R and L = OX (1) is the very ample line bundle on X and I is a homogeneous
ideal in the homogeneous coordinate ring of X, of finite colength.
In case I is the maximal ideal of R we denote HK multiplicity by eHK (R) or eHK (X, L).
In this talk we discuss HK multiplicity of two dimensional graded ring R (of projective curve)
over a field k of char p > 0, with respect to a homogeneous ideal I of finite colength. Without
loss of generality one can assume that the ring is a domain and k is algebraically closed. Let I
be generated by homogeneous generators f1 , . . . , fk of degrees d1 , . . . , dk respectively. Consider
the following short exact sequence of sheaves of OX -modules
0 −→ VL,I −→ ⊕i OX (1 − di ) −→ OX (1) 7→ 0,
where OX (1 − di ) → OX (1) is given by multiplication by fi . Then π ∗ (VL,I ) is a vector bundle
e where π : X
e → X is the normalization of X.
of rank k − 1 on X,
We express HK multiplicity in terms of (i) “standard” invariants of the curve which areconstant in a flat family and (ii) normalized slopes of the quotients occuring in a stronglye As a consequence
semistable Harder-Narasimhan filtration (HN filtration) of π ∗ (VL,I ) on X.
we getTheorem 1.1.
(1) eHK (R, I) is a rational number.
(2) Moreover if f1 , . . . , fk are minimal generators of I thenX
XeHK ≥ d/2((di )2 /(t − 1) −d2i )ii∗

and the equality holds if and only if V = π VX,L is strongly semistable, i.e., F s∗ (V )e where F s : X
e →X
e is the s-fold iterated Frobenius
is semistable, for all s ≥ 0, on X,
map.
1991 Mathematics Subject Classification. 13D40.
12V. TRIVEDI(The first part of the above theorem was also proved independently by Brenner, for generaldi and by the speaker for di = 1, giving essentially the same proof).
In case of plane curves, when p > d(d − 3), we have a simple numerical characterizationof semistablity of the Frobenius pullback of the kernel bundle VL,I , via HK multiplicity. In
particular we prove the following.
Theorem 1.2. Let X be an irreducible plane curve of degree d, i.e., X = Proj R = k[x, y, z]/(f ),where f is an irreducible homogeneous polynomial of degree d. Then, there exist integers l ands such that one of the following occurs:
3d(1) eHK (X, OX (1)) =43dl2(2) eHK (X, OX (1)) =, where s ≥ 1 and l ≡ pd(mod 2) with 0 < l ≤ d(d − 3).
+44dp2s
3dl2(3) eHK (X, OX (1)) =+ , where l ≡ d(mod 2) and 0 < l ≤ d.
44d(4) Moreover, if X is nonsingular then the only possible HK multiplicities are of type (1)or (2).
(5) If X has a singular point of multiplicity r ≥ d/2 theneHK (X, OX (1)) =3d (2r − d)2+.
44d

Examples computed so far show that eHK (R, I) can depend on char p of the ring, and anatural question is to ask how the HK multiplicities of reductions (mod p) of a given variety(in char. 0) vary with p. To make this more precise consider the following situation. Let k
be an algebraically closed field of characteristic 0. Let R be a finitely generated N-graded two
dimensional domain over k. Let I ⊂ R be an homogeneous ideal of finite colength. For thepair (R, I) we take a spread (A, RA , IA ), which means we choose a finitely generated Z-algebraA ⊆ k, a finitely generated N-graded algebra RA over A and an homogeneous ideal IA ⊂ RA
such that RA ⊗A k = R and for any closed point s ∈ Spec A (i.e. maximal ideal of A) the ringRs = RA ⊗A k(s) is a finitely generated N-graded 2-dimensional domain (which is a normal
domain if R is normal) over k(s) and the ideal Is = Im(IA ⊗A k(s)) ⊂ Rs is an homogeneous
ideal of finite colength.
We prove the following result.
Theorem 1.3. Let R be a standard graded two dimensional domain over k. Let I ⊂ R be an
homogeneous ideal of finite colength. Let (A, RA , IA ) be a spread as given above. Thenlim eHK (Rs , Is )s→s0exists, where s0 = Spec Q(A) is the generic point of Spec A, and the limit is taken over closed
points s ∈ SpecA.
The proof is again reduction to a problem on arbitrary vector-bundles over nonsingularprojective curves. We prove the following key lemma (which is not true for arbitrary p), making
crucial use of a result by Shepherd-Barron.
Lemma 1.4. If p ≥ 4(genus(X) − 1)(rank V )3 , then the HN filtration of F ∗ V is a refinementof the Frobenius pull back of the HN filtration of V ,We analyse the HN (Harder-Narasimahan) polygons HN Pps (V ) which corresponds to theiterated Frobenius pull back F s∗ V of vector bundle V on X. Using the above result we deducethat as p → ∞ the polygon
HN Pps (V ) → HN P (V ).
In particular the Hilbert-Kunz multiplicities of the reductions to positive characteristics ofan irreducible projective curve in characteristic 0 have a well-defined limit as the characteristictends to ∞. This limit, which is (relatively) an easier invariant to compute, is a lower boundHILBERT-KUNZ MULTIPLICITIES AND VECTOR BUNDLES OVER CURVES

3for the HK multiplicities of the reductions (mod p), though examples of Monsky show that theconvergence is not monotonic as p → ∞, in general.
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai400005, India
E-mail address: vija@math.tifr.res.in