The principal result of this paper is the theorem: Let C contained in Pn be a projectively normal curve over an algebraically closed field k. Let Y be the affine cone over C. Assume that H1(C, OC(1)) is nonzero. Then Y is not K0-regular. Moreover, when k=C is the field of complex numbers, then the cokernel of K0(Y) -> K0(Y x A1) is uncountably generated.
This theorem provides examples of varieties in characteristic
p which have only trivial vector bundles, but whose product
with the affine line acquires many non-trivial vector bundles. The
techniques of the proof rely heavily on the use of relative
K-theory and of relative cycle classes.
[3] Relative algebraic
K-theory
This paper contains careful proofs of certain ``folklore''
theorems describing the existence and basic properties of a relative
K-theory. The paper was written because of the nice
application found in the paper above, and to justify the use of the
relative theory there.
[4] A remark on K1 of an
algebraic surface
The principal result of this paper is the theorem: Let X be a surface over an algebraically closed field. Suppose that CH2(X) is finite dimensional in the sense of Mumford. Then the cokernel of the cup product map Pic(X) \otimesZ k* -> H1(X, K2) is N-torsion for some integer N. Moreover, if Alb(X) = 0, then there are isomorphisms Kn(k) \otimes Q -> H0(X, Kn) \otimes Q.
One application of this result is that, loosely speaking, any
isolated singularity which occurs on a Zariski surface cannot
contribute to the Chow group of an arbitrary normal surface.
[5] On SK1 of curves and
Kähler differentials
Krusemeyer studied SK1 of singular cubic curves in the affine plane by means of Mennicke symbols and ad hoc maps SK1(C) -> \Omega2k/Z. He asked if these results could be extended to nonsingular cubics. This paper shows
Theorem 1: Let C be a smooth curve contained in a smooth surface S over a field k. Then the image of SK1(C) in K0(S, C) (under the boundary map) is generated by the relative cycle classes of zero cycles on S-C which are relatively rationally equivalent to zero.
Theorem 2: If C is a smooth curve over
k, then the logarithmic derivative induces a map
SK1(C) -> H1(C,
K2) -> H1(C,
\Omega2C/Z).
When C is an affine cubic curve, these theorems imply
that SK1(C) is generated by classes of points of
the plane which do not lie on C, and that dlog
induces a homomorphism SK1(C) ->
\Omega2k/Z. This homomorphism can
be explicitly computed and is closely related to those of Krusemeyer.
[6] Zero cycles on del Pezzo surfaces over local
fields
Let X be a rational surface over a finite extension
L/Qp. This paper attempts to compute the
group A0(X) of rational equivalence classes of
zero cycles of degree zero on X. The first theorem is:
If X is a del Pezzo surface of degree d >
4, then A0(X) = 0. Next, a moduli
space is constructed for marked, split del Pezzo surfaces of degree
4. Cohomological considerations show that the only interesting
surfaces of degree 4 with quadratic splitting fields are Manin's
surfaces of Type IV. Using descent theory and the moduli space,
explicit equations are given for all Type IV surfaces. Finally,
A0(X) is explicitly computed for all Type IV surfaces
with unramified splitting field, by algebraic means, in terms of the
valuations in L of the coefficients of the equations.
[7] Hurwitz families and arithmetic Galois
groups
This paper is concerned with Galois branched covers of the Riemann
sphere. In the first section, we consider the moduli problem for such
covers, including whether Hurwitz families exist and whether they are
universal. In the second section, we turn to arithmetic questions
involving fields of definition and of moduli for covers and for
Hurwitz families. In the third section, we discuss some techniques for
constructing covers with given groups and field of definition, in
special cases.
[8] Inversion of abelian integrals on curves of
small genus
Let M be a smooth projective curve of genus g > 0. Let J be the Jacobian of M. Fix a base point P in M and define a map fi : M(i) -> J on the ith symmetric product of M by the formula fi(D) = D - iP. If i > 2g - 2, then fi is a Pi-g bundle. The inversion of abelian integrals problem asks: What are the transition functions of this bundle?
This paper presents a solution to this problem in the special
cases when g=2 or when g=3 and M is
not hyperelliptic. The techniques are very geometric but essentially
elementary.
[9] Motifs, L-functions, and the
K-cohomology of rational surfaces over finite
fields
Let X be a smooth projective surface over a finite
field F with q=pr elements. Let
Kj,X be the Zariski sheaf associated to
U |-> Kj(H0(U,
OU)). Let \phij be the
Frobenius endomorphism of Hj(X,
Ql). Set Pj(X,t) = det(1 -
\phijt) and Lj(X,s)=Pj(X,
q-s). The principal result of this paper is the
theorem: Let X be a rational surface over F
and let n > 2. Then, up to pth
powers,
L2i(X,n) = # Hi(X, K2n-i-1) L2i+1(X,n)= # Hi(X,
K2n-i-2)
This theorem is a special case of conjectures of Beilinson and
Lichtenbaum, simplified by the collapse of a spectral sequence.
The techniques rely on showing that the motif of X,
constructed from correspondences in the integral Chow theory, depends
only on the Galois representation on the Picard group. The
K-theory is then computed explicitly in terms of the
representation.
[10] On the K-theory of curves over finite
fields
The principal result of this paper computes the odd-dimensional
K-theory of a smooth projective curve over the algebraic
closure of a finite field. Two conjectures are then advanced
regarding the K-theory of fields, which would generalize known results
of Tate and Suslin. One of these conjectures was simultaneously and
independently made by Suslin. If these conjectures hold, then it is
shown that the K-theory of curves over finite fields can be
fully computed and will satisfy the Quillen-Lichtenbaum conjectures.
[11] Every rational surface is separably
split
The principal result of this paper is the theorem stated in the
title. This resolves a technical problem which was noticed when Bloch
introduced K-theory into the study of zero cycles on
rational surfaces. In addition, the paper contains the first complete
proof of the Adjunction Lemma in the form used by Iskovskih in his
classification of rational surfaces over arbitrary fields.
[12] Blood utilization during extracorporeal
membrane oxygenation
[13] On heterogeneous spaces
This paper presents a new method for finding all the rational
points on certain special curves. It contains numerous new examples
of curves of genus two and three to which the method is applied.
[14] The arithmetic of zero cycles on surfaces
with geometric genus and irregularity zero
[15] Attack priming and aggressive arousal in
female Syrian golden hamsters, Mesocritus auratus: Observed
attack latency and calculated attack probability as a function of
contact time with a conspecific
This paper is a presentation of the principal results of the
author's thesis.
[17] The arithmetic of Enriques
surfaces
This paper generalizes results from [?], [?], and [?] to the case of Enriques surfaces. Minimal models of Enriques surfaces over arbitrary fields are discussed. The K-theory of an Enriques surface over a finite field is computed up to 2- and p-torsion. Finally, the group of zero cycles of degree zero, modulo rational equivalence, on an Enriques surface with good reduction over a local field is computed, and this result is used to show that the corresponding group on an Enriques surface over a global field must be finite.
The first result of this paper uses the Beilinson-Gillet spectral sequence
in Deligne cohomology to describe the group of Albanese-trivial zero cycles
on any smooth projective variety, generalizing a result of Gillet. The paper
then explores the relations between zero cycles and integral differentials
of the third kind, leading to a new description of the group of zero cycles.
[22] Elliptic curves and logarithmic
derivatives
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