Let D>0 be the fundamental discriminant of a real quadratic field, and h(D) its class number. In this paper, by refining Ono's idea, we show that for any prime p>3, [sharp ]{0<D<X|h(D)[nequiv]0(mod p)}>>p√(X)/logX.

We study the moment map associated to the cotangent bundle of the space of representations of a quiver, determining when it is flat, and giving a stratification of its Marsden–Weinstein reductions. In order to do this we determine the possible dimension vectors of simple representations of deformed preprojective algebras. In an appendix we use deformed preprojective algebras to give a simple proof of much of Kac's Theorem on representations of quivers in characteristic zero.

In this paper we study the p-rank of Abelian prime-to-p covers of the generic r-pointed curve of genus g. There is an obvious bound on the p-rank of the cover. We show that it suffices to compute the p-rank of cyclic prime-to-p covers of the generic r-pointed curve of genus zero. In that situation, we show that, for large p, the p-rank of the cover is equal to the bound.

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that A[otimes]kk[x](x)/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x](x).

One is concerned with Cremona-like transformations, i.e., rational maps from $ P$n to $ P$m that are birational onto the image Y ⊂ $ P$m and, moreover, the inverse map from Y to $ P$n lifts to $ P$m. We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.