Study Of Algebraic Cycles Research Articles

The Tautological Ring of the Moduli Space M2,nrt

The purpose of this thesis is to study tautological rings of moduli spaces of curves. The moduli spaces of curves play an important role in algebraic geometry. The study of algebraic cycles on these spaces was started by Mumford. He introduced the notion of tautological classes on moduli spaces of curves. Faber and Pandharipande have proposed several deep conjectures about the structure of the tautological algebras. According to the Gorenstein conjectures these algebras satisfy a form of Poincare duality. The thesis contains three papers. In paper I we compute the tautological ring of the moduli space of stable n-pointed curves of genus one of compact type. We prove that it is a Gorenstein algebra. In paper II we consider the classical case of genus zero and its Chow ring. This ring was originally studied by Keel. He gives an inductive algorithm to compute the Chow ring of the space. Our new construction of the moduli space leads to a simpler presentation of the intersection ring and an explicit form of Keel’s inductive result. In paper III we study the tautological ring of the moduli space of stable n-pointed curves of genus two with rational tails. The Gorenstein conjecture is proved in this case as well.

An Elementary Proof of Suslin Reciprocity

AbstractWe state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor K2-groups (for function fields) employed in the proof.

Canonical quotient singularities in dimension three

We classify isolated canonical cyclic quotient singularities in dimension three, showing that, with two exceptions, they are all either Gorenstein or terminal. The proof uses the solution of a combinatorial problem which arose in the study of algebraic cycles on Fermat varieties.

Gersten's conjecture for arithmetic surfaces

One of the more promising applications of higher algebraic K-theory is to the study of algebraic cycles in algebraic geometry. By verifying a certain conjecture of Gersten for regular schemes of finite type over a field, Quillen has established a connection between the Chow group and certain sheaf cohomology groups defined in terms of K-theory [3]; Grayson has recently extended this result [2]. Their work indicates that verification of Gersten’s Conjecture might eventually prove to be the key step in developing intersection theory for regular schemes of unequal characteristic. For A a regular local ring, let &'(A) denote the (abelian) category consisting of those finitely generated A-modules M such that the support of M is of codimension at least p in Spec A. Gersten’s Conjecture is that the inclusion of categories JUPt’(A) -+ JGIP(A) induces the zero map of K-groups K,,(JH~“(A)) -+ K,(JUP(A)) for all n 2 0, p Z 0. In the paper in which he first stated the Conjecture, Gersten took the first step for regular local rings of unequal characteristic by proving the Conjecture for discrete valuation rings with finite residue class field [ 11. In this paper we take the next step by proving the Conjecture for the local rings of Spec R[t], where R is any Dedekind domain whose maximal ideals have finite residue class field; in particular, this applies when R is the ring of integers of an algebraic number field. The main theorem of this paper was first proved in the author’s 1976 doctoral dissertation at Rice University. The proof given here is basically the same, although the presentation has been greatly simplified by using the techniques developed in [5].