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© 2017 JETIR March 2017, Volume 4, Issue 3 www.jetir.org (ISSN-2349-5162) Generating Negative pedal curve through Inverse function – An Overview *Ramesha. H.G. Asst Professor of Mathematics. Govt First Grade College, Tiptur.

Abstract

This paper attempts to study the negative pedal of a curve with fixed point O is therefore the envelope of the lines perpendicular at the point M to the lines. In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·OQ = k2. The inverse of the curve C is then the locus of P as Q runs over C. The point O in this construction is called the center of inversion, the circle the circle of inversion, and k the radius of inversion.

An inversion applied twice is the identity transformation, so the inverse of an inverse curve with respect to the same circle is the original curve. Points on the circle of inversion are fixed by the inversion, so its inverse is itself. is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., f(x) = y if and only if g(y) = x. The inverse function of f is also denoted. Joseph-Louis Lagrange.The Lagrange inversion theorem (or Lagrange inversion formula, which we abbreviate as LIT), also known as the Lagrange--Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. The theorem was proved by Joseph-Louis Lagrange (1736--1813) and generalized by the German mathematician and teacher Hans Heinrich Bürmann ( --1817), both in the late 18th century. The Lagrange inversion formula is one of the fundamental formulas of combinatorics. In its simplest form it gives a formula for the power series coefficients of the solution f(x) of the function equation f(x)=xG(f(x)) f(x)=xG(f(x)) in terms of coefficients of powers of G.Not all functions have inverse functions. Those that do are called invertible. For a function f: X → Y to have an inverse, it must have the property that for every y in Y, there is exactly one x in X such that f(x) = y. This property ensures that a function g: Y → X exists with the necessary relationship with f. Using Riemann-Liouville fractional differential operator, a fractional extension of the Lagrange inversion theorem and related formulas are developed. The required basic definitions, lemmas, and theorems in the fractional calculus are presented.

Key words: differential operator, Lagrange inversion theorem, pedal, curve.

Introduction

A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. Then, a fractional version of Lagrange's expansion in more than one unknown function is generalized.

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The pedal of a curve with respect to a point is the locus of the foot of the perpendicular from to the tangent to the curve. More precisely, given a curve , the pedal curve of with respect to a fixed point (called the pedal point) is the locus of the point of intersection of the perpendicular from to a tangent to . The parametric equations for a curve relative to the pedal point are given by

(1)

(2)

If a curve is the pedal curve of a curve , then is the negative pedal curve of (Lawrence 1972, pp. 47- 48).

When a closed curve rolls on a straight line, the area between the line and roulette after a complete revolution by any point on the curve is twice the area of the pedal curve (taken with respect to the generating point) of the rolling curve. As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely, take a number x, multiply it by 5, then subtract 7 from the result), to reverse this and get x back from some output value, say y, we would undo each step in reverse order. In this case, it means to add 7 to y, and then divide the result by 5. In functional notation, this inverse function would be given by,

{\displaystyle g(y)={\frac {y+7}{5}}.}With y = 5x − 7 we have that f(x) = y and g(y) = x.

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The pedal curve of an astroid

(1)

(2)

with pedal point at the center is the quadrifolium

(3)

(4)

Objective:

This paper intends to explore pedal and negative pedal as inverse concepts. Also, to find Negative pedal of a curve C that can be defined as a curve C' such that the pedal of C is C'. Stating negative pedal curve is a plane curve that can be constructed from another plane curve C and a fixed point P on that curve.

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For an ellipse with parametric equations

(1)

(2)

the negative pedal curve with respect to the origin has parametric equations

(3)

(4)

(5)

(6)

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(7)

(8)

where

(9)

is the distance between the ellipse center and one of its foci and

(10)

is the eccentricity. For , the base curve is a circle, whose negative pedal curve with respect to the origin is also a circle. For , the curve becomes a "squashed" ellipse. For , the curve has four cusps and two ordinary double points and is known as Talbot's curve (Lockwood 1967, p. 157).

Taking the pedal point at a focus (i.e., ) gives the negative pedal curve

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(12)

Lockwood (1957) terms this family of curves Burleigh's ovals. As a function of the aspect ratio of an ellipse, the neagtive pedal curve varies in shape from a circle (at ) to an ovoid (for ) to a fish-shaped curve with a node and two cusps to a line plus a loop to a line plus a cusp.

The special case of the negative pedal curve for pedal point and (i.e., ) is here dubbed the fish curve.

Given a curve and a fixed point called the pedal point, then for a point on , draw a line perpendicular to . The envelope of these lines as describes the curve is the negative pedal of . It can be constructed by considering the perpendicular line segment for a curve parameterized by . Since one end of the perpendicular corresponds to the point , .

The equations of the negative pedal curve

Another end point can be found by taking the perpendicular to the line, giving

(1)

(2)

(3)

Plugging into the two-point form of a line then gives

(4)

(5)

Solving the simultaneous equations and then gives the equations of the negative pedal curve as

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(6)

(7)

If a curve is the pedal curve of a curve , then is the negative pedal curve of (Lawrence 1972, pp. 47- 48).

The following table summarizes the negative pedal curves for some common curves.

Curve pedal point negative pedal curve

cardioid negative pedal curve origin circle

cardioid negative pedal curve point opposite cusp cissoid of Diocles

circle negative pedal curve inside the circle ellipse

circle negative pedal curve outside the circle hyperbola

ellipse negative pedal curve with center Talbot's curve

ellipse negative pedal curve with focus ovoid

ellipse negative pedal curve with focus two-cusped curve

line any point parabola

parabola negative pedal curve origin semicubical parabola

parabola negative pedal curve focus Tschirnhausen cubic

Conclusion

A surface has negative curvature at a point if the surface curves away from the tangent plane in two different directions. The classic example is a saddle, which can be found on your body in the space between your thumb and forefinger, or along the inside of your neck. The concept of a surface of negative curvature can be generalized, for example, with respect to the dimension of the surface itself or the dimension and structure of the ambient space.

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© 2017 JETIR March 2017, Volume 4, Issue 3 www.jetir.org (ISSN-2349-5162) Surfaces of negative curvature locally have a saddle-like structure. This means that in a sufficiently small neighbourhood of any of its points, a surface of negative curvature resembles a saddle , not considering the behaviour of the surface outside the part of it that has been drawn). The local saddle-like character of the surface is clearly illustrated in this figure, which shows the principal sections of the surface at an arbitrary point OO. Let 1/R11/R1, 1/R21/R2 be their normal curvatures, i.e. the principal curvatures at the point OO( cf. Principal curvature). According to the classical definition, the Gaussian curvature at OO is the number K=1/R1R2K=1/R1R2. Since K<0K<0, the principal curvatures have different signs, for which reason the principal sections are convex in opposite directions; in Fig.1bthe section 0202 is convex in the direction of the normal nn, while the other is convex in the opposite direction, which fits in with the saddle-like character of the surface. The topological structure in the large ( "globally" ) of a surface of negative curvature can be very different.

References

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© 2017 JETIR March 2017, Volume 4, Issue 3 www.jetir.org (ISSN-2349-5162) 10. Rao, C.R. (1997) Statistics and Truth: Putting Chance to Work, World Scientific. ISBN 981-02-3111- 3 11. Rao, C.R. (1981). "Foreword". In Arthanari, T.S.; Dodge, Yadolah (eds.). Mathematical programming in statistics. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. pp. vii–viii. ISBN 978-0-471-08073-2. MR 0607328. 12. Whittle (1994, pp. 10–11, 14–18): Whittle, Peter (1994). "Almost home". In Kelly, F.P. (ed.). Probability, statistics and optimisation: A Tribute to Peter Whittle (previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28. ISBN 978-0-471-94829-2. Archived from the original on December 19, 2013. 13. Monastyrsky 2001, p. 1: "The Fields Medal is now indisputably the best known and most influential award in mathematics." 14. Riehm 2002, pp. 778–82. 15. Mathematics at the Encyclopædia Britannica 16. Benson, Donald C. (2000). The Moment of Proof: Mathematical Epiphanies. Oxford University Press. ISBN 978-0-19-513919-8. 17. Davis, Philip J.; Hersh, Reuben (1999). The Mathematical Experience (Reprint ed.). Mariner Books. ISBN 978-0-395-92968-1. 18. Gullberg, Jan (1997). Mathematics: From the Birth of Numbers (1st ed.). W. W. Norton & Company. ISBN 978-0-393-04002-9. 19. Hazewinkel, Michiel, ed. (2000). Encyclopaedia of Mathematics. Kluwer Academic Publishers. – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, and online. 20. Jourdain, Philip E. B. (2003). "The Nature of Mathematics". In James R. Newman (ed.). The World of Mathematics. Dover Publications. ISBN 978-0-486-43268-7. 21. Maier, Annaliese (1982). Steven Sargent (ed.). At the Threshold of Exact Science: Selected Writings of Annaliese Maier on Late Medieval Natural Philosophy. Philadelphia: University of Pennsylvania Press. 22. Perelman]. Interfax (in Russian). July 1, 2010. Retrieved 5 April 2015. Google Translated archived link at [1] (archived 2014-04-20) 23. Ritter, Malcolm (1 July 2010). "Russian mathematician rejects million prize". The Boston Globe. 24. Mackenzie, Dana (2006-12-22). "The Poincaré Conjecture—Proved". Science. 314 (5807): 1848– 1849. doi:10.1126/science.314.5807.1848. PMID 17185565. 25. Bing, R. H. (1958). "Necessary and sufficient conditions that a 3-manifold be S3". Annals of Mathematics. Second Series. 68 (1): 17–37. doi:10.2307/1970041. JSTOR 1970041. 26. Bing, R. H. (1964). "Some aspects of the topology of 3-manifolds related to the Poincaré conjecture". Lectures on Modern Mathematics. II. New York: Wiley. pp. 93–128.

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© 2017 JETIR March 2017, Volume 4, Issue 3 www.jetir.org (ISSN-2349-5162) 27. M., Halverson, Denise; Dušan, Repovš (23 December 2008). "The Bing–Borsuk and the Busemann conjectures". Mathematical Communications. 13 (2). arXiv:0811.0886. 28. Milnor, John (2004). "The Poincaré Conjecture 99 Years Later: A Progress Report" (PDF). Retrieved 2007-05-05. 29. Taubes, Gary (July 1987). "What happens when hubris meets nemesis". Discover. 8: 66–77. 30. Matthews, Robert (9 April 2002). "$1 million mathematical mystery "solved"". NewScientist.com. Retrieved 2007-05-05. 31. Szpiro, George (July 29, 2008). Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles. Plume. ISBN 978-0-452-28964-2. 32. Morgan, John W., Recent progress on the Poincaré conjecture and the classification of 3-manifolds. Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 1, 57–78 33. Hamilton, Richard (1982). "Three-manifolds with positive Ricci curvature". Journal of Differential Geometry. 17 (2): 255–306. doi:10.4310/jdg/1214436922. MR 0664497. Zbl 0504.53034. Reprinted in: Cao, H. D.; Chow, B.; Chu, S. C.; Yau, S.-T., eds. (2003). Collected Papers on Ricci Flow. Series in Geometry and Topology. 37. Somerville, MA: International Press. pp. 119–162. ISBN 1-57146- 110-8. 34. Perelman, Grigori (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159. 35. Perelman, Grigori (2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109. 36. Perelman, Grigori (2003). "Finite extinction time for the solutions to the Ricci flow on certain three- manifolds". arXiv:math.DG/0307245. 37. Kleiner, Bruce; John W. Lott (2008). "Notes on Perelman's Papers". Geometry and Topology. 12 (5): 2587–2855. arXiv:math.DG/0605667. doi:10.2140/gt.2008.12.2587. S2CID 119133773. 38. Cao, Huai-Dong; Xi-Ping Zhu (June 2006). "A Complete Proof of the Poincaré and Geometrization Conjectures – application of the Hamilton-Perelman theory of the Ricci flow" (PDF). Asian Journal of Mathematics. 10 (2). Archived from the original (PDF) on 2012-05-14.

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© 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) Enriques–Kodaira classification of compact complex surfaces – An Analysis *Ramesha. H.G. Asst Professor of Mathematics. Govt First Grade College, Tiptur.

Abstract

This paper attempts to study the Enriques–Kodaira classification a classification of compact complex surfaces into ten classes. the Enriques-Kodaira classification is a classification of compact complex surfaces. For complex projective surfaces it was done by Federigo Enriques, and Kunihiko Kodaira later extended it to non- algebraic compact surfaces. The classification of a collection of objects generally means that a list has been constructed with exactly one member from each isomorphism type among the objects, and that tools and techniques can effectively be used to identify any combinatorially given object with its unique representative in the list. Examples of mathematical objects which have been classified include the finite simple groups and 2- manifolds but not, for example, knots. A compact manifold is a manifold that is compact as a topological space. Examples are the circle (the only one-dimensional compact manifold) and the -dimensional sphere and torus. Compact manifolds in two dimensions are completely classified by their orientation and the number of holes (genus). It should be noted that the term "compact manifold" often implies "manifold without boundary," which is the sense in which it is used here. When there is need for a separate term, a compact boundaryless manifold is called a closed manifold.

It has been established that for given n the n-dimensional compact, connected complex manifolds X can be classified according to their Kodaira dimension kod(X), which can assume the values -00,0,1, ... , n. In the case n = 2 the surfaces in the classes kod(X) = -00 or kod(X) = 0, and to a lesser extent those with kod(X) = 1 can be classified in much more detail. Thus, starting from the rough classification by Kodaira dimension, surfaces are divided into ten classes. This classification is called the Enriques-Kodaira classification and is embodied in the following central result. The word "fibre space" will be used only for smooth surfaces fibred over smooth curves. Thus a fibre space is a triple (X , f , Y) , where X is a smooth surface, Y a smooth curve and f : X - Y a surjective morphism. The fibre space is called (relatively) minimal if no fibre contains an exceptional curve. It is called an elliptic fibre space if almost all fibres are elliptic curves. Sometimes a surface admitting at least one such elliptic fibration is called an elliptic surface. Algebraic surfaces X into four classes, mentioned in section ~, is the division according to the value of Kod(X) , which can be - - , 0 , 1 or 2 (we don’t need the interpretation given for the last two classes in section 1).

Key words: Elliptic Curve, Elliptic Surface, Effective Divisor, Kodaira Dimension.

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An Enriques surface is a smooth compact complex surface having irregularity and nontrivial canonical sheaf such that (Endraß). Such surfaces cannot be embedded in projective three-space, but there nonetheless exist transformations onto singular surfaces in projective three-space. There exists a family of such transformed surfaces of degree six which passes through each edge of a tetrahedron twice. A subfamily with tetrahedral symmetry is given by the two-parameter ( ) family of surfaces. For many problems in topology and geometry, it is convenient to study compact manifolds because of their "nice" behavior. Among the properties making compact manifolds "nice" are the fact that they can be covered by finitely many coordinate charts, and that any continuous real-valued function is bounded on a compact manifold.

For any positive integer , a distinct nonorientable surface can be produced by replacing disks with Möbius strips. In particular, replacing one disk with a Möbius strip produces a cross surface and replacing two disks produces the Klein bottle. The sphere, the -holed tori, and this sequence of nonorientable surfaces form a complete list of compact, boundaryless two-dimensional manifolds.

and the polynomial is a sphere with radius ,

(Endraß).

There are a large number of invariants that are linear combinations of the Hodge numbers, as follows:

* "b"0,"b"1,"b"2,"b"3,"b"4 are the Betti numbers: "b"i = dim("H"i("S")). "b"0 = "b"4 = 1 and "b"1 = "b"3 = "h""1","0" + "h""0","1" = "h""2","1" + "h""1","2" and "b"2 = "h""2","0" + "h""1","1" + "h""0","2"

of the trivial bundle. (It should not be confused with the Euler number. Add minus signs to the rows for dimensions 1 and 3. This is the sum along any side of the diamond, while the topological Euler-Poincaré characteristic is the sum over the whole diamond.) By Noether's formula it is also equal to the Todd genus ("c"12 +"c"2)/12 j "i","j" *τ is the signature (of the second cohomology group) and is equal to 4χ−"e", which is Σi,j(−1) "h" . *"b"+ and "b"− are the dimensions of the maximal positive and negative definite subspaces of "H"2, so "b"+ + − "b" ="b"2 and

"b"+ −"b"− =τ. 2 2 *"c"2 = "e" and "c"1 = "K" = 12χ − "e" are

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Friedman and Morgan proved that the invariants above depend only on the underlying smooth 4-manifold.

Objective:

This paper intends to explore classification that goes under the name of Enriques- Kodaira birational classification of compact complex surfaces. For each of these classes, the surfaces in the class can be parametrized by a moduli space

Blow-ups and birational maps

Let be a surface. The structure of rational maps between and any projective variety is simple by the following

Theorem 1 (Elimination of indeterminacy) Let be a rational map. Then there exists a composite of finite many blow-ups and a morphism such that .

Proof We have a bijection between nondegenerate rational maps and linear systems on of dimension which has no fixed components. We use this correspondence and induct on the dimension of the linear systems. Each blow-up drops the dimension and this process terminates using the intersection pairing. □

The structure of birational morphisms between surfaces is also rather simple. Suppose is a birational morphism of surfaces. Then can be decomposed as a sequence of blow-ups , and an isomorphism . Combining this fact with the elimination of indeterminacy, we obtain that

Theorem 2 Let be a birational map between surfaces. Then there exists morphisms and such that , where each of can be decomposed as a sequence of blow-ups and an isomorphism.

Neron-Severi groups and minimal models

The exponential exact sequence

induces a long exact sequence Hodge theory shows that is a lattice in , so the Picard variety is a complex torus (and indeed an abelian variety). Since , we obtain an exact sequence where is the image of .

Definition 1 The group is called the Neron-Severi group of .

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Since is finitely generated, the Neron-Severi group is also finitely generated. The rank of is called the Picard number of . The Neron-Severi group can also be described as the group of divisors on modulo algebraic equivalence ([3, V 1.7]).

The behavior of the Neron-Severi group under a blow-up can be easily seen.

Theorem 3 Let be a blow-up at a point with exceptional divisor . Then there is a canonical isomorphism .

Proof The canonical isomorphism given by descends to the Neron-Severi groups. □

In other words, a blow-up increases the Picard number by 1. Thus the Neron-Severi group gives us a canonical order on the set of isomorphism classes of surfaces birationally equivalent to a given surface .

Definition 2 Let . We say that dominates if there exists a birationally morphism (in particular, ). We say that is minimal if every birational morphism is an isomorphism.

In particular, every surface dominates a minimal surface since the Picard number is finite. Conversely, every surface is obtained by a sequence of blow-ups of a minimal surface. So the problem of birational classification boils down to classifying minimal surfaces.

We can characterize minimal surfaces as those without exceptional curves. By the very definition of blow-ups, an exceptional curve is isomorphic to and has self-intersection number . This is actually a useful numerical criterion of minimality of surfaces ([1, II. 17]).

Theorem 4 (Castelnuovo's contractibility criterion) Suppose a curve is isomorphic to with . Then is an exceptional curve on .

Ruled surfaces

In most cases, has a unique minimal element. However, the situation is not that simple for a large class of surfaces — ruled surfaces.

Definition 3 A surface is called ruled if it is birational to for a smooth projective curve.

As a little digression, we can calculate several birational invariants for ruled surfaces easily.

Definition 4 Let be any surface. We define

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 the irregularity . It is equal to by Hodge theory. It is the dimension of the Picard variety and the Albanese variety of . It is also the difference of the geometric genus and the arithmetic genus, hence its name.

 the geometric genus . It is equal to by Serre duality and by Hodge theory.

 the plurigenus ( ).

Theorem 5 Let be a ruled surface over . Then , and for all .

Proof Since these are birational invariants, we may assume . Using the isomorphism , we know that . Using the Künneth formula , we know that for all . □

Now let us step back to the problem of finding the minimal models of ruled surfaces. This is closely related to the notion of geometrically ruled surfaces.

Definition 5 A geometrically ruled surface over is a surface together with a smooth morphism whose fibers are isomorphic to .

The first thing to notice is that geometrically ruled surfaces form a subclass of ruled surfaces due to the following theorem ([1, III.4]).

Theorem 6 (Noether-Enriques) Let be a surface together with a smooth morphism . If for some point , is smooth over and is isomorphic to . Then there exists an open neighborhood of such that is a trivial bundle. In particular, every geometrically ruled surface is ruled.

Also from the Noether-Enriques theorem, we know that every geometrically ruled surface over admits local trivializations as a -bundle, thus they are are classified by the cohomology group . The exact sequence gives a long exact sequence

Since , classifies rank 2 vector bundle over and , one knows that every geometrically ruled surface over is actually -isomorphic to for some rank two vector bundle over . Moreover, such and are -isomorphic if and only if for some line bundle over .

Now let us see how geometrically ruled surfaces play a significant role in classifying minimal ruled surfaces.

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Proof Suppose is a minimal surface and is a birational map. Then by elimination of indeterminacy, we can find another surface fitting in the following diagram

where is the projection onto the first factor and is a composition of blow-ups. Suppose is the smallest such integer. If , let be the exceptional curve of the -th blow-up, then must be a point since is irrational by assumption. So we can eliminate the -th blow-up, which contradicts the minimality of . Therefore and is actually a morphism with its generic fiber isomorphic to . Hence ([1, III.8]) is a geometrically ruled surface over . □

In other words, for an irrational ruled surface, its minimal models are not unique and are classified by those projective bundles : the theory of rank two vector bundles over a curve is delicate, but more or less understood.

Rational surfaces and Castelnuovo's theorem

The ruled surfaces with base curve are called Hirzebruch surfaces. In particular, they are rational surfaces. Among these, the only geometrically ruled ones are ( ) since every vector bundle over is a direct sum of line bundles. The above classification of minimal models of ruled surfaces fails for Hirzebruch surfaces. A calculation of intersection numbers on ruled surface implies the following result ([1, IV.1]).

Proposition 1 If , then there is a unique irreducible curve on with negative self-intersection. Moreover, its self-intersection is .

It follows that 's are distinct and minimal for . However, it also follows that there is an irreducible curve on with . Hence by the uniqueness, coincides with the blow-up of at one point, hence is not minimal.

In order to find minimal models for rational surfaces, we need the following nontrivial fact ([1, V.6]). Notice that any rational surface satisfies ( ).

Lemma 1 Let be a minimal surface with . Then there exists a smooth rational curve on such that .

Now we can deduce the following classification of minimal models of rational surfaces.

Theorem 8 The minimal rational surfaces are the Hirzebruch surfaces ( ) and itself.

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© 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) Proof Let be a minimal rational surfaces. By the lemma, there exists smooth curves on with the least nonnegative . Choose such a curve with the least , where is a hyperplane section of . Then using the minimality and Riemann-Roch, one can show that every divisor is a smooth rational curve. Since the linear system of curves of passing through with multiplicity has codimension in . We know that . Suppose , then for any , the exact sequence implies that has no base point and as . Therefore . When , the morphism is geometrically ruled over , hence is for some . When , each fiber of the morphism is the intersection of two distinct rational curves, hence a point. Therefore . □

A similar argument using above useful lemma implies the following numerical characterization of rational surfaces ([1, V.1]).

Theorem 9 (Castelnuovo's Rationality Criterion) Let be a surface with . Then is rational.

Castelnuovo's Rationality Criterion together with the usage of Albanese varieties will enable us to finally show the uniqueness of the minimal models of all non-ruled surfaces ([1, V.19]).

Theorem 10 Let be two minimal non-ruled surfaces. Then every birational map between and is an isomorphism. In particular, every non-ruled surface admits a unique minimal model.

Hence we have found a complete list of minimal surfaces by now.

Kodaira dimension

Castelnuovo's Rationality Criterion provides a handy numerical tool to distinguish rational surfaces from others. We would like to see how the birational invariants will help us classifying surfaces. This can be achieved for ruled surfaces as well ([1, VI.18]).

Theorem 11 (Enriques) Let be a surfaces with (or ). Then is ruled. In particular, for all .

In view of the important role played by the plurigenera, we introduce the notion of Kodaira dimension for a smooth projective variety.

Definition 6 Let be a smooth projective variety and be the rational map from to the projective space associated to the complete linear system . We define the Kodaira dimension of to be the maximal dimension of the images of for . We write if for .

In particular, the Kodaira dimension is always no greater than the dimension of . Some examples are in order.

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The Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.

For the 9 classes of surfaces other than general type, there is a fairly complete description of what all the surfaces look like (which for class VII depends on the global spherical shell conjecture, still unproved in 2009). For surfaces of general type not much is known about their explicit classification, though many examples have been found.

The classification of algebraic surfaces in positive characteristic (Mumford 1969, Mumford & Bombieri 1976, 1977) is similar to that of algebraic surfaces in characteristic 0, except that there are no Kodaira surfaces or surfaces of type VII, and there are some extra families of Enriques surfaces in characteristic 2, and hyperelliptic surfaces in characteristics 2 and 3, and in Kodaira dimension 1 in characteristics 2 and 3 one also allows quasielliptic fibrations. These extra families can be understood as follows: In characteristic 0 these surfaces are the quotients of surfaces by finite groups, but in finite characteristics it is also possible to take quotients by finite group schemes that are not étale.

References

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© 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) 31. Einstein, Albert (1923). Sidelights on Relativity: I. Ether and relativity. II. Geometry and experience (translated by G.B. Jeffery, D.Sc., and W. Perrett, Ph.D). E.P. Dutton & Co., New York. 32. Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 33. Blass, Andreas (1984), "Existence of bases implies the axiom of choice", Axiomatic set theory (Boulder, Colorado, 1983), Contemporary Mathematics, 31, Providence, R.I.: American Mathematical Society, pp. 31–33, MR 0763890 34. Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247- 8419-5 35. Lang, Serge (1987), Linear algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96412-6 36. Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 37. Mac Lane, Saunders (1999), Algebra (3rd ed.), pp. 193–222, ISBN 978-0-8218-1646-2 38. Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871- 454-8 39. Roman, Steven (2005), Advanced Linear Algebra, Graduate Texts in Mathematics, 135 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-24766-3 40. Spindler, Karlheinz (1993), Abstract Algebra with Applications: Volume 1: Vector spaces and groups, CRC, ISBN 978-0-8247-9144-5 41. van der Waerden, Bartel Leendert (1993), Algebra (in German) (9th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56799-8

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