Let a, b, c be linearly independent homogeneous polynomials in the standard Z-graded ring R := k[s, t] with the same degree d and no common divisors. This defines a morphism P 1 → P 2 . The Rees algebra Rees(I ) = R⊕I ⊕I 2 ⊕· · · of the ideal I = a, b, c is the graded R-algebra which can be described as the image of an R-algebra homomorphism h: R[x, y, z] → Rees(I ). This paper discusses one result concerning the structure of the kernel of the map h and its relation to the problem of finding the implicit equation of the image of the map given by a, b, c. In particular, we prove a conjecture of Hong, Simis and Vasconcelos. We also relate our results to the theory of adjoint curves and prove a special case of a conjecture of Cox. Published by Elsevier B.V.

We define the concept of regularity for bigraded modules over a bigraded polynomial ring. In this setting we prove analogs of some of the classical results on m-regularity for graded modules over polynomial algebras. * We would like to thank William Adkins and David Cox for numerous discussions and suggestions. Brought to you by | University of Michigan Authenticated Download Date | 6/21/15 2:17 AM 3. R is 0-regular. This follows from Serre's calculations of the cohomology of the invertible sheaves OðkÞ on P n ([18]), as reinterpreted by Grothendieck in the language of local cohomology (combine [8, Proposition (2.1.5)] with [9, Exp. II, Proposition 5]).

Let S be a parametrized surface in P 3 given as the image of φ : P 1 × P 1 → P 3 . This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of S when certain base points are present. This work extends the algorithm provided by Cox [5] for when φ has no base points, and it is analogous to some of the results of Busé, Cox, and D'Andrea [2] for the case when φ : P 2 → P 3 has base points.

Abstract. Let A2(n) denote the quotient of the Siegel upper half space of degree two by Γ2(n), the principal congruence subgroup of level n in Sp(4, Z). A2(n) is the moduli space of principally polarized abelian varieties of dimension two with a level n structure, and has a compactification A2(n) * first constructed by Igusa. When n ≥ 3 this is a smooth projective algebraic variety of dimension three.In this work we analyze the topology of A2 (3) * and the open subset A2(3). In this way we obtain the rational cohomology ring of Γ2(3). The key is that one has an explicit description of A2 (3) * : it is the resolution of the 45 nodes on a projective quartic threefold whose equation was first written down about 100 years ago by H. Burkhardt. We are able to compute the zeta function of this variety reduced modulo certain primes.

A weak version of the Ihara formula is proved for zeta functions attached to quotients of the Bruhat-Tits building of PGL 3 . This formula expresses the zeta function in terms of Hecke-Operators. It is the first step towards an arithmetical interpretation of the combinatorially defined zeta function.