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Algebraic geometry and analytic geometry

In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties. Background Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way. For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem). For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant. Thus f is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere. Important results There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century and still continuing today. Some of the more important advances are listed here in chronological order. Riemann's existence theorem Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves — that is, such coverings all come from finite extensions of the function field. The Lefschetz principle In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0, by treating K as if it were the complex number field. It roughly asserts that true statements in algebraic geometry over C are true over any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski and are based in mathematical logic. This principle permits the carrying over of results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0. Chow's theorem Chow's theorem, proved by W. L. Chow. is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective space that is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased concisely as "any analytic subspace of complex projective space which is closed in the strong topology is closed in the Zariski topology." This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry. Serre's GAGA Foundations for the many relations between the two theories were put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Serre, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves. Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings. Formal statement of GAGA Let (X,\mathcal O_X) be a scheme of finite type over C. Then there is a topological space Xan which as a set consists of the closed points of X with a continuous inclusion map λX: Xan → X. The topology on Xan is called the "complex topology" (and is very different from the subspace topology). Suppose φ: X → Y is a morphism of schemes of locally finite type over C. Then there exists a continuous map φan: Xan → Yan such λY ° φan = φ ° λY. There is a sheaf \mathcal O_X^{an} on Xan such that (X^{an}, \mathcal O_X^{an}) is a ringed space and λX: Xan → X becomes a map of ringed spaces. The space (X^{an}, \mathcal O_X^{an}) is called the "analytifiction" of (X,\mathcal O_X) and is an analytic space. For every φ: X → Y the map φan defined above is a mapping of analytic spaces. Furthermore, the map φ ↦ φan maps open immersions into open immersions. If X = C[x1,...,xn] then Xan = Cn and \mathcal O_X^{an}(U) for every polydisc U is a suitable quotient of the space of holomorphic functions on U. For every sheaf \mathcal F on X (called algebraic sheaf) there is a sheaf \mathcal F^{an} on Xan (called analytic sheaf) and a map of sheaves of \mathcal O_X -modules \lambda_X^*: \mathcal F\rightarrow (\lambda_X)_* \mathcal F^{an} . The sheaf \mathcal F^{an} is defined as \lambda_X^{-1} \mathcal F \otimes_{\lambda_X^{-1} \mathcal O_X} \mathcal O_X^{an} . The correspondence \mathcal F \mapsto \mathcal F^{an} defines an exact functor from the category of sheaves over (X, \mathcal O_X) to the category of sheaves of (X^{an}, \mathcal O_X^{an}) . The following two statements are the heart of Serre's GAGA theorem (as extended by Grothendieck, Neeman et al.) If f: X → Y is an arbitrary morphism of schemes of finite type over C and \mathcal F is coherent then the natural map (f_* \mathcal F)^{an}\rightarrow f_*^{an} \mathcal F^{an} is injective. If f is proper then this map is an isomorphism. One also has isomorphisms of all higher direct image sheaves (R^i f_* \mathcal F)^{an} \cong R^i f_*^{an} \mathcal F^{an} in this case. Now assume that Xan is hausdorff and compact. If \mathcal F, \mathcal G are two coherent algebraic sheaves on (X, \mathcal O_X) and if f: \mathcal F^{an} \rightarrow \mathcal G^{an} is a map of sheaves of \mathcal O_X^{an} modules then there exists a unique map of sheaves of \mathcal O_X modules \varphi: \mathcal F\rightarrow \mathcal G with f = φan. If \mathcal R is a coherent analytic sheaf of \mathcal O_X^{an} modules over Xan then there exists a coherent algebraic sheaf \mathcal F of \mathcal O_X -modules and an isomorphism \mathcal F^{an} \cong \mathcal R . Moishezon manifolds A Moishezon manifold M is a compact connected complex manifold such that the field of meromorphic functions on M has transcendence degree equal to the complex dimension of M. Complex algebraic varieties have this property, but the converse is not (quite) true. The converse is true in the setting of algebraic spaces. In 1967, Boris Moishezon showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric.

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, in a given dimension, and that geometric transformations are permitted that move the extra points (called "points at infinity") to traditional points, and vice versa. The properties that are meaningful in projective geometry are those that are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations); the first issue for geometers is what kind of geometric language would be adequate to the novel situation. It is not possible to talk about angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions.

While the ideas were available earlier, projective geometry was mainly a development of the nineteenth century. A huge body of research made it the most representative field of geometry of that time. This was the theory of complex projective space, since the coordinates used (homogeneous coordinates) were complex numbers. Several major strands of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme leading to the study of the classical groups) built on projective geometry. It was also a subject with a large number of practitioners for its own sake, under the banner of synthetic geometry. Another field that emerged from axiomatic studies of projective geometry is finite geometry.

The field of projective geometry is itself now divided into many research subfields, two examples of which are projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations).

Overview

Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines. That there is indeed some geometric interest in this sparse setting was seen as projective geometry was developed by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes (that always meet), and other linear subspaces, which exhibit the principle of duality. The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. the line through them) and "two distinct lines determine a unique point" (i.e. their point of intersection) show the same structure as propositions. Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy. It was realised that the theorems that do hold in projective geometry are simpler statements. For example the different conic sections are all equivalent in (complex) projective geometry, and some theorems about circles can be seen as special cases of these general theorems.

In the early 19th century the work of Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics . Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri

Pyramid (geometry)

This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural pyramids, seePyramid (disambiguation).

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base.

A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.

When unspecified, the base is usually assumed to be square.

If the base is a regular polygon and the apex is above the center of the polygon, an n-gonal pyramid will have Cnv symmetry.

Pyramids are a subclass of the prismatoids.

Pyramids with regular polygon faces

The regulartetrahedron, one of the Platonic solids, is a triangular pyramid all of whose faces are equilateral triangles. Besides the triangular pyramid, only the square and pentagonal pyramids can be composed of regular convex polygons, in which case they are Johnson solids.

Star pyramids

Pyramids with regular star polygon bases are called star pyramids. For example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides.

Volume

The volume of a pyramid is V= \tfrac{1}{3}Bh where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base. In 499 CE Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya(section 2.6) .

The formula can be formally proved using calculus: By similarity, the dimensions of a cross section parallel to the base increase linearly from the apex to the base. Then, the cross section at any height y is the base scaled by a factor of 1 - \tfrac{y}{h}, where h is the height from the base to the apex. Since the area of any shape is multiplied by the square of the shape's scaling factor, the area of a cross section at height y is \frac{B(h - y)^2}{h^2}. The volume is given by the integral

\frac{B}{h^2} \int_0^h (h-y)^2 \, dy = \frac{-B}{3h^2} (h-y)^3 \bigg|_0^h = \tfrac{1}{3}Bh.

The volume of a pyramid whose base is a regular n-sided polygon with side length s and whose height is h is therefore:

V = \frac{n}{12}hs^2 \cot\frac{\pi}{n}.

The volume of a pyramid whose base is a regular n-sided polygon with radius R is therefore:

V = \frac{nR^2h}{6} \sin{\frac{2\pi}{n}}.

This property can be used to derive the volume for cones, as well. See also cone.

Surface area

The surface area of a pyramid is A= B + \frac{PL}{2} where B is the base area, P is the base perimeter and L is the slant height: L= \sqrt{h^2+r^2} where h is the pyramid altitude and r is the inradius of the base.


Birational geometry

In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian school of algebraic geometry in the years 1890–1910. From about 1970 advances have been made in higher dimensions, giving a good theory of birational geometry for dimension three.

Birational geometry is largely a geometry of transformations, but it doesn't fit exactly with the Erlangen programme. One reason is that its nature is to deal with transformations that are only defined on an open, dense subset of an algebraic variety. Such transformations, given by rational functions in the co-ordinates, can be undefined not just at isolated points on curves, but on entire curves on a surface, and so on.

A formal definition of birational mapping from one algebraic variety V to another is that it is a rational mapping with a rational inverse mapping. This has to be understood in the extended sense that the composition, in either order, is only in fact defined on a non-empty Zariski open subset.

One of the first results in the subject is the birational isomorphism of the projective plane, and a non-singular quadricQ in projective 3-space. Already in this example whole sets have ill-defined mappings: taking a point P on Q as origin, we can use lines through P, intersecting Q at one other point, to project to a plane — but this definition breaks down with all lines tangent to Q at P, which in a certain sense 'blow up' P into the intersection of the tangent plane with the plane to which we project.

That is, quite generally, birational mappings act like relations, with graphs containing parts that are not functional. On an open dense set they do behave like functions, but the Zariski closures of their graphs are more complex correspondences on the product showing 'blowing up' and 'blowing down'. Detailed descriptions of those, in terms of projective spaces associated to tangent spaces can be given and justified by the theory.

An example is the Cremona group of birational automorphisms of the projective plane. In purely algebraic terms, for a given fieldK, this is the automorphism group over K of the field K(X, Y) of rational functions in two variables. Its structure has been analysed since the nineteenth century, but it is 'large' (while the corresponding group for the projective line consists only of Möbius transformations determined by three parameters). It is still the subject of research.



From Yahoo Answers

Question:find as many examples of geometry as possible: like parallel and perpendicular lines; types of angles, different polygons, circles and semicircles, 3 dimensional shapes such as prisms, pryramid, cylinders, cones, domes, or spheres, symmetry including reflections, rotations, and translations. thankss! much help appreciated!! =]

Answers:Each base corner is a square, and these support another larger square base. The base of the tower is much larger than the top, and the width decreases with height for purposes of strength, forming a triangular type of shape. In terms of mechanics of materials, the architecture use triangles as a shape because it is so statically determinate, which means that it can carry great loads and also has rigidity. From the sides, use is made of circular and parabolic arches, again to add strength. The intricate structure of the tower is lattice columns in which diagonals connect at points which brace and make rigid lightweight columns. There are many rows of rectangular columns adding even more support and strength. Here is a good site which explains in more detail the architecture in terms of all the geometry and angles: http://www.ce.jhu.edu/perspectives/studies/Eiffel%20Tower%20Files/ET_Geometry.htm

Question:find as many examples of geometry as possible: like parallel and perpendicular lines; types of angles, different polygons, circles and semicircles, 3 dimensional shapes such as prisms, pryramid, cylinders, cones, domes, or spheres, symmetry including reflections, rotations, and translations. thankss! much help appreciated!! =]

Answers:all of the above

Question:Parallel lines with a non-perpendicular transversal Complementary adjacent angles formed by two intersecting lines Concentric triangles with parallel edges Internally tangent circles Trapezoid with two right angles 3-4-5 triangle Two non-perpendicular bisecting line segments andand.. is a star a concave polygon? O_o Also, is pacman a good example of a major arc not part of a full circle? Thank you!

Answers:If you weld bars of metal together in lines. I think a star is a concave polygon. Yes pacman is a good example

Question:Determine the most complete classification guaranteed for each quadrilateral described. Drawing counter examples will help you eliminate incorrect classifications. 1. An art student uses strips of paper to form a quadrilateral with congruent diagonals. This quadrilateral must be a parallelogram a rectangle an isosceles trapezoid of no guaranteed classification 2. On a blueprint for a house, if the master bedroom is a parallelogram with at least one right angle, then the master bedroom will be the shape of a rectangle a rhombus a square just a parallelogram 3. If a framer makes a quadrilateral frame out of four congruent boards, no matter how it is put together this frame will always be a square a rhombus a rectangle of no guaranteed classification 4. A quadrilateral foundation with four congruent angles must be the shape of a square a parallelogram a rectangle no guaranteed classification 5. If a game board is a quadrilateral with diagonals that bisect each other, the game board will be a square a parallelogram a rectangle of no guaranteed classification 6. While putting sheeting on the outside of a house, a construction worker cuts a piece of plywood such that it is a quadrilateral with exactly two consecutive right angles. This plywood has the shape of a square a rectangle a trapezoid a parallelogram 7. The instructions of an art project tell students to cut a piece of paper into a quadrilateral with two pairs of congruent sides. If all of the students follow the instructions, each paper will be a rectangle a parallelogram an isosceles trapezoid of no guaranteed classification

Answers:1) rectangle 2) rectangle 3) rhombus 4) rectangle 5) parallelogram 6) trapezoid 7) of no guaranteed classification

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