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A pair of symmetric bilinear forms A and B determine a binary form $f(x,y) = disc(Ax-By)$. We prove that the question of whether a given binary form can be written in this way as a discriminant form generically satisfies a local-global principle and deduce from this that most binary forms over $\mathbf{Q}$ are discriminant forms. This is related to the arithmetic of the hyperelliptic curve $z^2 = f(x,y)$. Analogous results for non-hyperelliptic curves are also given., Comment: v3: updated references

Let P n \textbf {P}^n be the projective n n -space over the complex numbers. In this note we show that an indecomposable rigid coherent sheaf on P n \textbf {P}^n has a trivial endomorphism algebra. This generalizes a result of Drézet for n = 2. n=2.

This paper attempts to characterize the dicritical set of a rational function which generates a special pencil at a simple point of an algebraic or arithmetical surface.

For a dominant algebraically stable rational self-map of the complex projective plane of degree at least 2, we will consider three different definitions of the Fatou set and show the equivalence of them (Ascoli-Arzela type theorem). As a corollary, it follows that all Fatou components are Stein. This is an improvement of an early result by Fornaess and Sibony.

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension., Comment: Accepted for publication in Transactions AMS, submitted October 8, 2003

In (6), the geometric dimension of all stable vector bundles over real projective spaceP n was determined ifn is even and sufficiently large with respect to the order 2 e of the bundle in f

We study subvarieties of a general projective degree d d hypersurface X d ⊂ P n X_d\subset \mathbf {P}^n . Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the following sharp corollary: any subvariety of a general hypersurface X d ⊂ P n X_{d}\subset {\mathbf P}^n , for n ≥ 6 n\geq 6 and d ≥ 2 n − 2 d\geq 2n-2 , is of general type.

For a coanalytic–complete or \wideutilde {\Pi }^1_2–complete subspace X X of a Polish space we prove that there exists a continuous bijection of X X onto the Hilbert cube [ 0 , 1 ] N [0,1]^{\mathbb {N}} . This extends results of Pytkeev. As an application of our main theorem we give an answer to some questions of Arkhangelskii and Christensen. Under the assumption of Projective Determinacy we also give some generalizations of these results to higher projective classes.

We prove the existence of a quasi-projective moduli scheme for principal bundles over an arbitrary projective scheme.

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the Ext \operatorname {Ext} -algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the “no loop” conjecture.

Given six lines in P 3 \mathbf {P}^3 , quartics through the six lines define a map from P 3 \mathbf {P}^3 to P 4 \mathbf {P}^4 , and the image of this map is described in terms of invariants of the six lines. The map can be interpreted as projection of the six lines, and this permits a description of the canonical model of the octic surface which is given by points which project the lines so that they are tangent to a conic. We also define polarity for sets of six lines, and discuss the above map in the case of a self-polar set of lines and in the case of six lines which form a “double-sixer” on a cubic surface.

In this paper, we study the bicanonical pencil of a Godeaux surface and of a determinantal Barlow surface. This study gives a simple proof for the unobstructedness of deformations of a determinantal Barlow surface. Then we compute the number of hyperelliptic curves in the bicanonical pencil of a determinantal Barlow surface via classical Prym theory.

The present paper is devoted to the classification of irregular surfaces of general type with p g ≥ 3 p_{g}\geq 3 and nonbirational bicanonical map. Our main result is that, if S S is such a surface and if S S is minimal with no pencil of curves of genus 2 2 , then S S is the symmetric product of a curve of genus 3 3 , and therefore p g = q = 3 p_{g}=q=3 and K 2 = 6 K^{2}=6 . Furthermore we obtain some results towards the classification of minimal surfaces with p g = q = 3 p_{g}=q=3 . Such surfaces have 6 ≤ K 2 ≤ 9 6\leq K^{2}\leq 9 , and we show that K 2 = 6 K^{2}=6 if and only if S S is the symmetric product of a curve of genus 3 3 . We also classify the minimal surfaces with p g = q = 3 p_{g}=q=3 with a pencil of curves of genus 2 2 , proving in particular that for those one has K 2 = 8 K^{2}=8 .

Burstall classified conformal non-superminimal harmonic two-tori in spheres and complex projective spaces. In this paper, we shall classify conformal non-superminimal harmonic two-tori in a 2or 3-dimensional quaternionic protective space, which are not always covered by primitive harmonic two-tori of finite type.

In order to tile the unit square with rational triangles, at least four triangles are needed. There are four candidate configurations: one is conjectured not to exist; two others are dealt with elsewhere; the fourth is the "nu-configuration," corresponding to rational points on a quartic surface in affine 3-space. This surface is examined via a pencil of elliptic curves. One rank-3 curve is treated in detail, and rational points are given on 772 curves of the pencil. Within the range of the search there are roughly equal numbers of odd and even rank and those of rank 2 or more seem to be at least 0.45 times as numerous as those of rank 0. Symmetrical solutions correspond to rational points on a curve of rank 1, which exhibits an almost periodic behavior.

Local variational min-sup characterizations are presented for the real spectrum of a selfadjoint operator pencil. Instead of minimizing over all subspaces of fixed codimension as in the classical result, the new characterizations minimize over subspaces that are close to extremal subspaces. In this way, the entire real spectrum, including continuous spectrum, can be characterized.

Fix integers n , k , d n,k,d with n ≥ 2 , d ≥ 2 n\ge 2,d\ge 2 and k > 0 k>0 ; if n = 2 n=2 assume d ≥ 3 d\ge 3 . Let P 1 , … , P k P_1,\dotsc ,P_k be general points of the complex projective space P n \mathbf {P}^n and let π : X → P n \pi :X\to \mathbf {P}^n be the blow up of P n \mathbf {P}^n at P 1 , … , P k P_1,\dotsc ,P_k with exceptional divisors E i := π − 1 ( P i ) E_i:=\pi ^{-1}(P_i) , 1 ≤ i ≤ k 1\le i\le k . Set H := π ∗ ( O P n ( 1 ) ) H:=\pi ^*(\mathbf {O}_{\mathbf {P}^n}(1)) . Here we prove that the divisor L := d H − ∑ 1 ≤ i ≤ k E i L:=dH-\sum _{1\le i\le k}E_i is ample if and only if L n > 0 L^n>0 , i.e. if and only if d n > k d^n>k .

We find the maximum number of orthogonal skewsymmetric anticommuting integer matrices of order n for each natural number n and relate this to finding free direct summands of certain generic projective modules. While studying composition of quadratic forms, Hurwitz [4] and Radon [6] considered families of orthogonal matrices {A1,3L , A,} satisfying the conditions (1) Ai= -At, i= 1, ,s (2) AiAj = -AjAi, i $ j. DEFINITION. (1) A family of orthogonal matrices satisfying (1) and (2) above will be called a Hurwitz-Radon (H-R)family. If n is a positive integer and n=2ab, b odd, then we write a=4c+d where O0d

In previous papers we developed a version of homological algebra for Z [ G ] Z[G] modules ( G G a finite group) using summands of permutation modules in place of projective modules. The resulting theory is more discriminating than the usual homological algebra. For example, an f.g. Z Z -torsion free Z [ G ] Z[G] module is either projective or has infinite projective dimension, whereas for G G cyclic, all f.g. Z [ G ] Z[G] modules have permutation projective dimension one. In this paper we apply results of Endo and Miyata on permutation projective modules to characterize groups of dimension two.

The minimum number of triple points of a substantial generic immersion of a projection plane in a lens space of type ( 2 k , q ) (2k,q) depends quadratically on k k .

R. H. Bruck has pointed out the one-to-one correspondence between the isomorphism classes of certain translation planes, called subregular, and the equivalence classes of disjoint circles in a finite miquelian inversive plane I P ( q ) IP(q) . The problem of determining the number of isomorphism classes of translation planes is old and difficult. Let q be an odd prime-power. In this paper, a study of sets of disjoint circles in I P ( q ) IP(q) enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order q 2 {q^2} which are subregular of index 3 or 4. It is conjectured (and proved for n ⩽ 3 n \leqslant 3 ) that, given a set of n disjoint circles in I P ( q ) IP(q) , the numbers of circles disjoint from each of the given n circles is asymptotic to q 3 / 2 n {q^3}/{2^n} . This conjecture, if true, would allow one to estimate the number of subregular translation planes of order q 2 {q^2} with any positive index.

If V is a complex analytic subvariety of pure dimension k in the unit ball in C n {{\mathbf {C}}^n} which does not contain the origin, then the 2k-volume of V equals the measure computed with multiplicity of the set of ( n − k ) (n - k) -complex subspaces through the origin which meet V. The measure of this set computed without multiplicity is a smaller quantity which is nevertheless bounded below by a number depending only on the distance from V to the origin. As an application we characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. The complex analysis which we shall need will be developed in the context of uniform algebras.

Equivalence problems for abstract, and induced, projective structures are investigated. (i) The notion of induced projective structures on submanifolds of a projective space is rigorously defined. (ii) Equivalence problems for such structures are discussed; in particular, it is shown that nonplanar surfaces in R P 3 \mathbf {R}{P^3} are all projectively equivalent to each other. (iii) The imbedding problem of abstract projective structures is studied; in particular, we show that every abstract projective structure on a 2 2 -manifold arises as an induced structure on an arbitrary nonplanar surface in R P 3 \mathbf {R}{P^3} ; this result should be contrasted to that of Chern (see [6]).

Simple formulas in terms of Chern classes are given for the volume of a tube about a Kähler submanifold of a space of constant holomorphic sectional curvature. A comparison theorem which generalizes these formulas is also given. Tubes about complete intersections in complex projective space are studied in detail.

The purpose of this paper is to find and discuss the involutorial transformations belonging to the special complex of lines which meet a rational curve r of order m, and having a pencil of invariant surfaces j Fj I which contain the curve as an (n 2)-fold basis element. If n = 2 the curve r?f is not a basis curve. A pencil of quadrics and any rational curve always lead to a result contained among those found by Montesano. t The remaining admissible cases are as follows

1. Statement of the problem. It has been shown by Enriquest that if an algebraic surface contains a pencil of elliptic curves and also two curves cutting each curve of the pencil in two groups of points which are not equivalent and which do not possess equivalent multiples, the surface is invariant under a discontinuous infinite series of birational transformations. In a more recent memoir: the same author states that, with the exception of the case in which

A net of curves on a surface consists of two one-parameter families of curves such that through each point of the portion of the surface under consideration there passes just one curve of each family, the two tangents at each point being distinct. It is a characteristic property of conjugate nets that at every surface point the two tangents of every conjugate net separate the tangents of the asymptotic net harmonically. In other words, at every point of a surface the cross ratio of the two asymptotic tangents and any two conjugate tangents is -1. Recent developments in projective differential geometry seem to indicate that these ideas should be generalized. In the first place, the number -1 may be replaced by any number r, and we may then consider the class of all nets on a surface, every one of which has the property that at every surface point its tangents form with the asymptotic tangents the constant cross ratio r. In the second place, the asymptotic net may be replaced by an arbitrary fundamental net. We are thus led to introduce the following definition. The class of all nets on a surface, every one of which has the property that at every surface point its two tangents form with the two tangents of a fundamental net the same cross ratio, will be called a bundle of nets. A bundle of nets on a surface is therefore determined by a fundamental net and a number r. For example, all conjugate nets on a surface constitute a bundle, for which the fundamental net is the asymptotic net, and for which r =-1. The one-parameter family of conjugate nets on a surface, each of which has the property that, at every surface point, its tangents form with the tangents of a fundamental conjugate net a constant cross ratio, has been calledt by Wilczynski a pencil of conjugate nets. This definition can be extended from the bundle of conjugate nets to any bundle of nets. A pencil of nets in a bundle is a one-parameter family of nets in the bundle, each of which has the property that, at every surface point, its two tangents form with the two tangents of a fundamental net of the bundle a constant cross ratio. This cross ratio is different for different nets of the pencil, but for a given

Proposed and compared are four equivalent sets R, S, T, D of self-dual axioms for projective geometries, using points, hyperplanes and incidence as primitive elements and relation. The set R is inductive on the number of dimensions. The sets S, T, D all include the axiom “on every n points there is a plane", the dual of this axiom, one axiom on the existence of a certain configuration, and one or several axioms on the impossibility of certain configurations. These configurations consist of ( n + 1 ) (n + 1) points and ( n + 1 ) (n + 1) planes for sets S, T, but of ( n + 2 ) (n + 2) points and ( n + 2 ) (n + 2) planes for set D. Partial results are obtained by a preliminary study of self-dual axioms for simplicial spaces (spaces which may have fewer than 3 points per line).

1. SCHROETER'S t classic work on the general cubic leaves little to be desired in point of symmetry and generality. It is nevertheless interesting to build up the theory of the unicursal cubic, the curve being defined as the locus of the intersection of corresponding rays of two projective pencils, one of the first and the other of the second order. This has in fact been done by DRASCH. t The following discussion, based likewise on this definition, is materially simplified by the use of the properties of the point designated in ? 7 by E. Incidentally the investigation brings to light a remarkable one-to-one correspoindence between the points of the plane and the line elements on the cubic. 2. The locus described above has at least one and at inost three points in common with any line in the plane. We assume the truth of this theorem, a proof of which may be found in the eleventh chapter of REYE's Geometrie der Lage. Notations.-Throughout this paper we shall use the following notations: The pencil of the first order will be denoted by s, its center by S, and its rays by a, 6, c, etc. The pencil of the second order (and also the conic enveloped by it) will be denoted by K, and the rays by a, /3, 'y, etc. The cubic itself will be denoted by C. 3. THEOREM.-NO point of the cubic C lies within the conic K. 4. THEOREM.-The cubic C touches the conic K in at least one point, and at most in three. To prove this take S', a point on K, for the center of a pencil s' of the first order perspective to K. This pencil generates with s a conic which cuts KC in at least one and at most three other points besides S'. These are easily seen to be points on C.

is the Schwarzian derivative and the coefficienits are single-valued functions on a given algebraic configurationi; and he has given the solution for the hyperelliptic case, and the case of a canonical Riemain surface. He raised the question of what the form would be in the case of a plane non-singular quartic, p = 3, considered in the projective plane, when one imposes the fuLrther condition that the answer shall be given in terms of invariant expressions which bear symmetrically on the three ternary homogeneous variables x1, x2, x3, and the questioni was answered by GordanJ+ This last restriction, though doubtless initeresting, is not prescribed by the nature of the problem, which admits an altogether satisfactory projective treatment for the case that one assumes the given algebraic configuration in the form of Noether's normal C.-,. Let the projective homog,eneous coordinates of a point of this curve be denoted by (x1, x., X), and let the curve be projected on a pencil of hyperplanes