# Cayley-Bacharach theorems and conjectures

@article{Eisenbud1996CayleyBacharachTA, title={Cayley-Bacharach theorems and conjectures}, author={D. Eisenbud and M. Green and Joe W. Harris}, journal={Bulletin of the American Mathematical Society}, year={1996}, volume={33}, pages={295-324} }

A theorem of Pappus of Alexandria, proved in the fourth century A.D., began a long development in algebraic geometry. In its changing expressions one can see reflected the changing concerns of the field, from synthetic geometry to projective plane curves to Riemann surfaces to the modern development of schemes and duality. We survey this development historically and use it to motivate a brief treatment of a part of duality theory. We then explain one of the modern developments arising from it… Expand

#### 219 Citations

A Cayley-Bacharach theorem and plane configurations

- Mathematics
- 2021

In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley–Bacharach condition. In particular, by bounding the number of points satisfying the… Expand

Cayley-Bacharach theorems with excess vanishing

- Mathematics
- 2019

Griffiths and Harris showed in 1978 that if E is a rank n vector bundle on a smooth projective variety of dimension n, and if s is a section of E vanishing simply on a finite set Z, then any section… Expand

On the uniformity of zero-dimensional complete intersections

- Mathematics
- 2013

Abstract After showing that the General Cayley–Bacharach Conjecture formulated by D. Eisenbud, M. Green, and J. Harris (1996) [6] is equivalent to a conjecture about the region of uniformity of a… Expand

The Projective Geometry of the Gale Transform

- Mathematics
- 1998

Abstract The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises and in subjects as diverse as optimization, coding theory, theta functions, and… Expand

Finite subschemes of abelian varieties and the Schottky problem

- Mathematics
- 2010

The Castelnuovo-Schottky theorem of Pareschi-Popa characterizes Jacobians, among indecomposable principally polarized abelian varieties of dimension g, by the existence of g+2 points in general… Expand

Uniqueness of Steiner laws on cubic curves

- Mathematics
- 2006

In this paper we use the Cayley-Bacharach theorem of classical algebraic geometry to construct several universal algebras on algebraic curves using divisors and complete intersection cycles and study… Expand

3264 and All That: A Second Course in Algebraic Geometry

- Mathematics
- 2016

Introduction 1. Introducing the Chow ring 2. First examples 3. Introduction to Grassmannians and lines in P3 4. Grassmannians in general 5. Chern classes 6. Lines on hypersurfaces 7. Singular… Expand

Hilbert Series and Lefschetz Properties of Dimension One Almost Complete Intersections

- Mathematics
- 2014

We generalize some results about the graded Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal J of dimension… Expand

The Penrose transform and solutions of partial differential equations

- Mathematics
- 2019

We study the origins of twistor theory and of the Penrose twistor, from the point of view of partial differential equations. We show how the fundamental ideas require the ability to correctly… Expand

On the Cayley-Bacharach Property

- Mathematics
- 2018

Abstract The Cayley-Bacharach Property (CBP), which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

Cayley-Bacharach schemes and their canonical modules

- Mathematics
- 1993

A set of s points in P d is called a Cayley-Bacharach scheme (CB-scheme), if every subset of s − 1 points has the same Hilbert function. We investigate the consequences of this «weak uniformity.» The… Expand

Gorenstein algebras and the Cayley-Bacharach theorem

- Mathematics
- 1985

This paper is an examination of the connection between the classical Cayley-Bacharach theorem for complete intersections in p2 and properties of graded Gorenstein algebras. Introduction. It is known,… Expand

Commutative Algebra: with a View Toward Algebraic Geometry

- Mathematics
- 1995

Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a… Expand

Mathematical Thought from Ancient to Modern Times

- Mathematics
- 1972

This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creation… Expand

A Source Book in Mathematics, 1200-1800

- History
- 1969

These selected mathematical writings cover the years when the foundations were laid for the theory of numbers, analytic geometry, and the calculus.Originally published in 1986.The Princeton Legacy… Expand

A Source Book in Mathematics

- Nature
- 1930

THIS is a very entertaining volume, a surprisingly successful attempt to do what nearly all good judges would have declared to be impossible. Its aim is “to present the most significant passages from… Expand

The History of Mathematics

- Medicine
- Nature
- 1889

THE quaint words addressed “to the great variety of readers” by the editors of the folio Shakespeare of 1623 are equally applicable to the useful compendium of mathematical history which is the… Expand

Projective Geometry

- Nature
- 1911

IN the first page of their introduction the authors say:Projective Geometry.By Prof. O. Veblen Prof. J. W. Young. Vol. i. Pp. x + 342. (Boston and London: Ginn and Co., 1910.) Price 15s. net.

Algebraic Geometry

- Nature
- 1973

Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)

The Algebraic Theory of Modular Systems

- Mathematics, Computer Science
- 1916

Introduction 1. The resultant 2. General properties of modules 3. The inverse system 4. Expand