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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.

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Question:definitions to thesee geometry terms.? -angle -adjacent angles -verticanl angles -complementary angles -supplementary angles -intersection -parallel lines -ray you don't need to give all, just some will be ok. but please they have to be amazingly accurate! =]

Answers:An angle is an amount of turning Adjacent angles represent the angles on a straight line and add up to 180 degrees Vertical angles ? ? ? Complementary angles add up to 90 degrees Supplementary angles add up to 180 degrees Intersection is the point at which 2 lines cross Parallel lines lie in the same plane and they never meet, no matter how far they are produced (made longer) Ray is one arm of an angle. Hope this helps, Twiggy.

Question:I have to do this geometry dictionary & thats the only one i have left. Cant find anywhere on the internet, & the algebra teacher wants it tomorrow.

Answers:When I hear legs i think of the legs of right triangle, where they are the two non-hypotenuse sides joined by a right angle. hope that helps.

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:PROJECT DETAILS: Name of Project: ABC s of Geometry You will create a book using vocabulary words that you have learned this year. To do so you must choose a geometry term for each letter of the alphabet. For example, angle for A or endpoint for E. You will have one letter per page of your book (your book should be 26 pages). For each page you will: 1. State the letter of the alphabet (in order). 2. Give the geometry term and a brief definition. 3. Provide and illustration of something that represents/contains/describes the term. This can be a picture from a magazine or something you have drawn by hand. Your projects will be graded on completeness (using all letters of the alphabet with all items for each), correctness (correct definition and pictures used), neatness, and creativity. Your projects should be completely colorful. You should also have a cover for your book. i have all the letters, definition, and pictures.. All i need help with is what can i do to make it creative? And what would be a good/creative title?

Answers:To make it creative maybe you could some sort of theme for your examples/pictures, like winter wonderland, or under the sea, or if you have a hobby/interest like astronomy or ancient Rome or something like that you could use that as you theme idk its just a thought. and for a title, maybe something like: - Blocked Letters (I know its kinda lame but its the best i could come up with. lol sry)