Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques from commutative algebra, for solving geometrical problems about these sets of zeros; the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, parabolas, hyperbolas, cubic curves like elliptic curves, quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.

Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.

A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way, similar to its use in the study of differential and analytic manifolds; this is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring.

This means that a point of such a scheme may be either a subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed.

We consider the affine space of dimension n over denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the

Cambridge University Press

Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world, it holds letters patent as the Queen's Printer. The press mission is "to further the University's mission by disseminating knowledge in the pursuit of education and research at the highest international levels of excellence". Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global sales presence, publishing hubs, offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries, its publishing includes academic journals, reference works and English language teaching and learning publications. Cambridge University Press is a charitable enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press.

It originated from letters patent granted to the University of Cambridge by Henry VIII in 1534, has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses. Authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, Stephen Hawking. University printing began in Cambridge when the first practising University Printer, Thomas Thomas, set up a printing house on the site of what became the Senate House lawn – a few yards from where the press's bookshop now stands. In those days, the Stationers' Company in London jealously guarded its monopoly of printing, which explains the delay between the date of the university's letters patent and the printing of the first book. In 1591, Thomas's successor, John Legate, printed the first Cambridge Bible, an octavo edition of the popular Geneva Bible; the London Stationers objected strenuously. The university's response was to point out the provision in its charter to print "all manner of books".

Thus began the press's tradition of publishing the Bible, a tradition that has endured for over four centuries, beginning with the Geneva Bible, continuing with the Authorized Version, the Revised Version, the New English Bible and the Revised English Bible. The restrictions and compromises forced upon Cambridge by the dispute with the London Stationers did not come to an end until the scholar Richard Bentley was given the power to set up a'new-style press' in 1696. In July 1697 the Duke of Somerset made a loan of £200 to the university "towards the printing house and presse" and James Halman, Registrary of the University, lent £100 for the same purpose, it was in Bentley's time, in 1698, that a body of senior scholars was appointed to be responsible to the university for the press's affairs. The Press Syndicate's publishing committee still meets and its role still includes the review and approval of the press's planned output. John Baskerville became University Printer in the mid-eighteenth century.

Baskerville's concern was the production of the finest possible books using his own type-design and printing techniques. Baskerville wrote, "The importance of the work demands all my attention. Caxton would have found nothing to surprise him if he had walked into the press's printing house in the eighteenth century: all the type was still being set by hand. A technological breakthrough was badly needed, it came when Lord Stanhope perfected the making of stereotype plates; this involved making a mould of the whole surface of a page of type and casting plates from that mould. The press was the first to use this technique, in 1805 produced the technically successful and much-reprinted Cambridge Stereotype Bible. By the 1850s the press was using steam-powered machine presses, employing two to three hundred people, occupying several buildings in the Silver Street and Mill Lane area, including the one that the press still occupies, the Pitt Building, built for the press and in honour of William Pitt the Younger.

Under the stewardship of C. J. Clay, University Printer from 1854 to 1882, the press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks. During Clay's administration, the press undertook a sizeable co-publishing venture with Oxford: the Revised Version of the Bible, begun in 1870 and completed in 1885, it was in this period as well that the Syndics of the press turned down what became the Oxford English Dictionary—a proposal for, brought to Cambridge by James Murray before he turned to Oxford. The appointment of R. T. Wright as Secretary of the Press Syndicate in 1892 marked the beginning of the press's development as a modern publishing business with a defined editorial policy and administrative structure, it was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories. The Cambridge Modern History was published

Projective space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when V = R2 and V = R3 are the real projective line and the real projective plane where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, R3 denotes ordered triplets of real numbers; the idea of a projective space relates to perspective, more to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the vector space is R3 with the camera entrance pupil at the origin, the projective space corresponds to the image points. Projective spaces can be studied as a separate field in mathematics, but are used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates.

As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard Euclidean geometry for the plane, two lines always intersect at a point except when parallel. In a projective representation of lines and points, such an intersection point exists for parallel lines, it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, their representation theories; as outlined above, projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the real projective plane P2 in some detail. There are three equivalent definitions: The set of all lines in R3 passing through the origin.

Every such line meets the sphere of radius one centered in the origin twice, say in P = and its antipodal point. P2 can be described as the points on the sphere S2, where every point P and its antipodal point are not distinguished. For example, the point is identified with, etc, yet another equivalent definition is the set of equivalence classes of R3 ∖, i.e. 3-space without the origin, where two points P = and P∗ = are equivalent iff there is a nonzero real number λ such that P = λ⋅P∗, i.e. x = λx∗, y = λy∗, z = λz∗. The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point in R3, is. The last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with z ≠ 0 is equivalent to. So there are two disjoint subsets of the projective plane: that consisting of the points = for z ≠ 0, that consisting of the remaining points; the latter set can be subdivided into two disjoint subsets, with points and. In the last case, x is nonzero, because the origin was not part of P2.

This last point is equivalent to. Geometrically, the first subset, isomorphic to R2, is in the image the yellow upper hemisphere, or equivalently the lower hemisphere; the second subset, isomorphic to R1, corresponds to the green line, or, equivalently the light green line. We have the red point or the equivalent light red point. We thus have a disjoint decomposition P2 = R2 ⊔ R1 ⊔ point. Intuitively, made precise below, R1 ⊔ point is itself the real projective line P1. Considered as a subset of P2, it is called line at infinity, whereas R2 ⊂ P2 is called affine plane, i.e. just the usual plane. The next objective is to make the saying "parallel lines meet at infinity" precise. A natural bijection between the plane z = 1 and the sphere of the projective plane is accomplished by the gnomonic projection; each point P on this plane is mapped to the two intersection points of the sphere with the line through its center and P. These two points are identified in the projective plane. Lines in the plane are mapped to great circles if one includes one pair of antipodal points on the equator.

Any two great circles intersect in two antipodal points. Great circles corresponding to parallel lines intersect on the equator. So any two lines have one intersection point inside P2; this phenomenon is axiomatized in projective geometry. The real projective space of dimension n or projective n-space, Pn, is the set of the lines in Rn+1 passing through the origin. For defining it as a topological space and as an algebraic variety it is better to define it as the quotient space of Rn+1 by the equivalence relation "to be aligned with the origin". More Pn:= / ~,where ~ is the equivalence relation defined by: ~ if there is a non-zero real number λ such that =; the elements of the projective space are called points. The projective coordinates of a point P are x0... xn, where is any element of the corresponding equivalence class. This is denoted P =, the colons and the brackets emphasizing that the right-hand side is an equivalence class, whic

David Mumford

David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, for research into vision and pattern theory. He was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science, he is a University Professor Emeritus in the Division of Applied Mathematics at Brown University. Mumford was born in West Sussex in England, of an English father and American mother, his father William started an experimental school in Tanzania and worked for the newly created United Nations. In high school, he was a finalist in the prestigious Westinghouse Science Talent Search. After attending the Phillips Exeter Academy, Mumford went to Harvard, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1955 and 1956, he completed his Ph. D. in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques, he published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory, on the equations defining an abelian variety, on algebraic surfaces.

His books Abelian Varieties and Curves on an Algebraic Surface combined the new theories. His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique, the only accessible introduction, they are now available as The Red Book of Schemes. Other work, less written up were lectures on varieties defined by quadrics, a study of Goro Shimura's papers from the 1960s. Mumford's research did much to revive the classical theory of theta functions, by showing that its algebraic content was large, enough to support the main parts of the theory by reference to finite analogues of the Heisenberg group; this work on the equations defining abelian varieties appeared in 1966–7. He published some further books of lectures on the theory, he was one of the founders of the toroidal embedding theory. In a sequence of four papers published in the American Journal of Mathematics between 1961 and 1975, Mumford explored pathological behavior in algebraic geometry, that is, phenomena that would not arise if the world of algebraic geometry were as well-behaved as one might expect from looking at the simplest examples.

These pathologies fall into two types: bad behavior in characteristic p and bad behavior in moduli spaces. Mumford's philosophy in characteristic p was as follows: A nonsingular characteristic p variety is analogous to a general non-Kähler complex manifold. In the first Pathologies paper, Mumford finds an everywhere regular differential form on a smooth projective surface, not closed, shows that Hodge symmetry fails for classical Enriques surfaces in characteristic two; this second example is developed further in Mumford's third paper on classification of surfaces in characteristic p. This pathology can now be explained in terms of the Picard scheme of the surface, in particular, its failure to be a reduced scheme, a theme developed in Mumford's book "Lectures on Curves on an Algebraic Surface". Worse pathologies related to p-torsion in crystalline cohomology were explored by Luc Illusie. In the second Pathologies paper, Mumford gives a simple example of a surface in characteristic p where the geometric genus is non-zero, but the second Betti number is equal to the rank of the Néron–Severi group.

Further such examples arise in Zariski surface theory. He conjectures that the Kodaira vanishing theorem is false for surfaces in characteristic p. In the third paper, he gives an example of a normal surface; the first example of a smooth surface for which Kodaira vanishing fails was given by Michel Raynaud in 1978. In the second Pathologies paper, Mumford finds that the Hilbert scheme parametrizing space curves of degree 14 and genus 24 has a multiple component. In the fourth Pathologies paper, he finds reduced and irreducible complete curves which are not specializations of non-singular curves; these sorts of pathologies were considered to be scarce when they first appeared. But Ravi Vakil in a paper called "Murphy's law in algebraic geometry" has shown that Hilbert schemes of nice geometric objects can be arbitrarily "bad", with unlimited numbers of components and with arbitrarily large multiplicities. In three papers written between 1969 and 1976, Mumford extended the Enriques–Kodaira classification of smooth projective surfaces from the case of the complex ground field to the case of an algebraically closed ground field of characteristic p.

The final answer turns out to be as the answer in the complex case, once two important adjustments are made. The first is that one may get "non-classical" surfaces, which come about when p-torsion in the Picard scheme degenerates to a non-reduced group scheme; the second is the possibility of obtaining quasi-elliptic surfaces in characteristics three. These are surfaces fibred over a curve where the general fibre is a curve of arithmetic genus one with a cusp. Once these adjustments ar