Hilbert-Kunz multiplicity of plane curves and a conjecture of K. Pardue

grantor: University of TorontoIn this thesis I study problems related to the Hilbert-Kunz multiplicity of projective varieties, in particular projective plane curves. For a homogeneous form 'f' ∈'S' = ' k'['x'1,···,' k'n] of degree 'd', we define a ' GLn'('k') invariant w∈R , the Frobenius index, of 'f'. For curves defined by ' zd' - 'f' ('x, y'), we then can express the Hilbert-Kunz multiplicity as a quadratic polynomial in the Frobenius index of 'f' ('x, y') (Theorem 3.3.3). This enables us to determine explicitly the Hilbert-Kunz multiplicity in the case where 'f' ('x, y') has a factor of multiplicity at least 'd'/2 (Corollary 3.3.4). For a form f∈Z&sqbl0;x,y,z&sqbr0; of degree 'd' and content 1, the scheme Spec Z&sqbl0;x,y,&sqbr0;/f over Spec( Z ) has Q&sqbl0;x,y,z&sqbr0;/f as its generic fiber and Fp&sqbl0;x,y,z&sqbr0;/ f&d4; as the special fiber over a prime 'p' ( f&d4; is the reduction of 'f' over Fp ). The behavior of the Hilbert-Kunz multiplicity may be difficult to understand for some special fibers. But is there an intrinsic way to determine these invariants for "most" fibers? Along this line of thought we make a conjecture (Problem 4) for smooth curves and verify it for diagonal curves 'xd' + 'y d' + 'zd' by applying the Han-Monsky algorithm to them. In the last chapter we investigate numerical approaches to the Hilbert-Kunz multiplicity of plane curves. In particular following our work in [3] we prove a conjecture of K. Pardue (Conjecture 1). This yields a nice structure theorem for minimal free resolutions of the Gorenstein ideals ('xq', ' yq', 'zq') : ' f' with 'f' a smooth cubic (section 4.7). In fact in this case we are able to prove that the degrees of minimal generators of the ideal are given by a linear function in 'q'. Numerical experiments show that this is a general phenomenon and a general result along this line would settle at least the rationality problem for the Hilbert-Kunz multiplicity of plane curves.Ph.D