Projective spaces and Bézout s theorem

Transcription

1 Projective spaces and Bézout s theorem êaû{0 Mijia Lai 5 \ laimijia@sjtu.edu.cn

2 Outline 1. History 2. Projective spaces 3. Conics and cubics 4. Bézout s theorem and the resultant 5. Cayley-Bacharach theorem 6. Pappus s theorem and Pascal s theorem

3 Chronology of developments leading to algebraic geometry (300 B.C.) Greeks: Euclid, Apollonius,... study of conics I (Middle Age) Arabs,... algebra, good notation of algebra, (First half of 17th century) Fermat, Descartes,... analytic geometry, )ÛAÛ (Start of 19th century) Monge, Poncelet, Klein,... projective geometry, KAÛ

4 Chronology of developments leading to algebraic geometry (300 B.C.) Greeks: Euclid, Apollonius,... study of conics I (Middle Age) Arabs,... algebra, good notation of algebra, (First half of 17th century) Fermat, Descartes,... analytic geometry, )ÛAÛ (Start of 19th century) Monge, Poncelet, Klein,... projective geometry, KAÛ

5 Chronology of developments leading to algebraic geometry (300 B.C.) Greeks: Euclid, Apollonius,... study of conics I (Middle Age) Arabs,... algebra, good notation of algebra, (First half of 17th century) Fermat, Descartes,... analytic geometry, )ÛAÛ (Start of 19th century) Monge, Poncelet, Klein,... projective geometry, KAÛ

6 Chronology of developments leading to algebraic geometry (300 B.C.) Greeks: Euclid, Apollonius,... study of conics I (Middle Age) Arabs,... algebra, good notation of algebra, (First half of 17th century) Fermat, Descartes,... analytic geometry, )ÛAÛ (Start of 19th century) Monge, Poncelet, Klein,... projective geometry, KAÛ

7 Main theme of algebraic geometry (Second half of 19th century) Riemann, Plücker, Hilbert, Noether,... Algebraic geometry, êaû Study of the solutions of algebraic equations. The study is naturally connected with commutative algebra, complex analysis, differential geometry, number theory, projective geometry, topology, etc.

8 Main theme of algebraic geometry (Second half of 19th century) Riemann, Plücker, Hilbert, Noether,... Algebraic geometry, êaû Study of the solutions of algebraic equations. The study is naturally connected with commutative algebra, complex analysis, differential geometry, number theory, projective geometry, topology, etc.

9 Main theme of algebraic geometry (Second half of 19th century) Riemann, Plücker, Hilbert, Noether,... Algebraic geometry, êaû Study of the solutions of algebraic equations. The study is naturally connected with commutative algebra, complex analysis, differential geometry, number theory, projective geometry, topology, etc.

10 Plane curves in Cartesian coordinates Lines are of the form ax + by + c = 0.

11 Plane curves in Cartesian coordinates Lines are of the form ax + by + c = 0. Ellipse x 2 a 2 + y 2 b 2 = 1.

12 Plane curves in Cartesian coordinates Lines are of the form ax + by + c = 0. Ellipse Parabola x 2 a 2 + y 2 b 2 = 1. y = ax 2.

13 Plane curves in Cartesian coordinates Lines are of the form ax + by + c = 0. Ellipse Parabola Hyperbola x 2 a 2 + y 2 b 2 = 1. y = ax 2. x 2 a 2 y 2 b 2 = 1.

14 Study from transformation-invariant point view One can choose coordinates freely, e.g., one makes a linear transformation x = ax + by, y = cx + dy. Algebraic equation changes, but the shape does not change. Motivated by Perspective in drawing, which was introduced by Italian Renaissance painters and architects, one could also freely move the plane in R 3 and consider projection of curve onto various planes. In doing so, one should add points at infinity to a plane to make the projection a bijection. This extended plane is called the projective plane RP 2.

15 Study from transformation-invariant point view One can choose coordinates freely, e.g., one makes a linear transformation x = ax + by, y = cx + dy. Algebraic equation changes, but the shape does not change. Motivated by Perspective in drawing, which was introduced by Italian Renaissance painters and architects, one could also freely move the plane in R 3 and consider projection of curve onto various planes. In doing so, one should add points at infinity to a plane to make the projection a bijection. This extended plane is called the projective plane RP 2.

16 Perspective

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18 Projection between two planes Light coming from O, there is no image for the line m P in Q, similarly, there is no pre-image for the line n Q in P. The solution is adding both planes a line at infinity.

19 Projective spaces A formal definition of RP 2 : R 3 \ (0, 0, 0)/, where (x, y, z) (x, y, z ) if there exists λ 0 such that (x, y, z) = λ(x, y, z ). It can be viewed as the set of all lines in R 3 passing through origin. We use the homogeneous coordinates [x : y : z] to denote a point in the projective plane, where x, y, z are not all zero. Projective plane can be viewed as the union of plane z = 1 and lines that are parallel to z = 1 ( points at infinity ).

20 Plane curve curve in the Projective plane Any polynomial g(x, y) R[x, y] of degree n can be homogenized by z n g( x z, y z ). Examples: g(x, y) = 3x + 5y 2 2, after homogenization, we get f (x, y, z) = 3xz + 5y 2 2z 2, a homogeneous polynomial in x, y, z. f (x, y, z) = 0 RP 2 g(x, y) = 0 {z = 1} points at infinity.

21 Plane curve curve in the Projective plane Any polynomial g(x, y) R[x, y] of degree n can be homogenized by z n g( x z, y z ). Examples: g(x, y) = 3x + 5y 2 2, after homogenization, we get f (x, y, z) = 3xz + 5y 2 2z 2, a homogeneous polynomial in x, y, z. f (x, y, z) = 0 RP 2 g(x, y) = 0 {z = 1} points at infinity.

22 Within this identification, we could study zeros of homogeneous polynomials in RP 2. They are usual plane curves with points at infinity added. Degree one (line): ax + by + cz = 0. Degree two (quadratic curve): ax 2 + by 2 + cz 2 + dxy + exz + fyz = 0. Degree three (cubic curve): ax 3 + by jxyz = 0. }{{} 10 terms

23 Within this identification, we could study zeros of homogeneous polynomials in RP 2. They are usual plane curves with points at infinity added. Degree one (line): ax + by + cz = 0. Degree two (quadratic curve): ax 2 + by 2 + cz 2 + dxy + exz + fyz = 0. Degree three (cubic curve): ax 3 + by jxyz = 0. }{{} 10 terms

24 Classification of quadratic curves in RP 2 x = ax + by + cz, y = dx + ey + fz, z = gx + hy + iz is called a linear transformation from RP 2 RP 2. Theorem Any curve of degree 2 in RP 2 can be transformed into one the following types: x 2 = 0, a double line; x 2 + y 2 = 0, a point; x 2 y 2 = 0, two lines, x 2 + y 2 + z 2 = 0, the empty set; x 2 + y 2 z 2 = 0, the unit circle (conic).

25 Classification of quadratic curves in RP 2 x = ax + by + cz, y = dx + ey + fz, z = gx + hy + iz is called a linear transformation from RP 2 RP 2. Theorem Any curve of degree 2 in RP 2 can be transformed into one the following types: x 2 = 0, a double line; x 2 + y 2 = 0, a point; x 2 y 2 = 0, two lines, x 2 + y 2 + z 2 = 0, the empty set; x 2 + y 2 z 2 = 0, the unit circle (conic).

26 Conics Ellipses, parabolas, hyperbolas are all same as circles. (They all arise as conic sections, i.e., intersection of a plane with a cone in various position, thus from projective point view they are the same) Conic sections

27 Cubics A cubic curve is irreducible if it cannot factor out degree 1 or 2 factors. Theorem Any irreducible cubic curve can be transformed into y 2 = x 3 + ax 2 + bx + c. In complex projective plane, such a curve looks like a torus, so it is called an elliptic curve.

28 Cubics A cubic curve is irreducible if it cannot factor out degree 1 or 2 factors. Theorem Any irreducible cubic curve can be transformed into y 2 = x 3 + ax 2 + bx + c. In complex projective plane, such a curve looks like a torus, so it is called an elliptic curve.

29 Cubics A cubic curve is irreducible if it cannot factor out degree 1 or 2 factors. Theorem Any irreducible cubic curve can be transformed into y 2 = x 3 + ax 2 + bx + c. In complex projective plane, such a curve looks like a torus, so it is called an elliptic curve.

30 Common zeros: solutions of algebraic systems Examples: 3x + 2y 1 = 0 and 3x + 2y 5 = 0 By homogenization, they become 3x + 2y z = 0 and 3x + 2y 5z = 0, by eliminating z, we get 3x + 2y = 0, and z = 0, thus they intersect at [ 2, 3, 0] RP 2. This is a point at infinity. So in projective plane, any two lines intersect at exactly one point. Example: x 2 y 2 = 1, x y = 0 By homogenization, they become x 2 y 2 = z 2 and x y = 0, it follows that z = 0, and x = y, thus they intersect at [1 : 1 : 0], but the intersection has a multiplicity 2.

31 Bézout s theorem Theorem (Bézout) Let f = 0 and g = 0 be two homogeneous polynomials in x, y, z variables of degree n and m respectively, assume they don t have any common component, then they intersect at most m n points in RP 2. Assumption as above, f and g intersect at exactly m n points, counting multiplicities, in the complex projective plane.

32 Resultant Given two polynomials f (x) = a n x n + + a 0 and g(x) = b m x m + + b 0, the resultant R(f, g) is defined to be R(f, g) = a m n b n m (αi β j ), where α i are roots of f (x), and β j are those of g(x). (By fundamental theorem of algebra, α i, β j exist as complex numbers) Proposition R(f, g) = 0 if and only if f and g have a common root.

33 In terms of coefficients, we have a n a n 1 a 0 a n a n 1 a 0 R(f, g) = a n a n 1 a 0 b m b 0 b m b 0 b m b 0

34 Determine a quadratic curve Any two points determine a line uniquely. How many points determine a quadratic curve uniquely? Heuristics: suppose the curve is ax 2 + bxy + cy 2 + dx + ey + f = 0, then plugging given points, we will get a system of linear equations on a, b, c, d, e, f - the six unknowns. Multiplying by a non-zero constant results in the same curve, so in general if given five points, we get five equations for six unknowns, from linear algebra, we know the solution set is at least one-dimensional.

35 Determine a quadratic curve Any two points determine a line uniquely. How many points determine a quadratic curve uniquely? Heuristics: suppose the curve is ax 2 + bxy + cy 2 + dx + ey + f = 0, then plugging given points, we will get a system of linear equations on a, b, c, d, e, f - the six unknowns. Multiplying by a non-zero constant results in the same curve, so in general if given five points, we get five equations for six unknowns, from linear algebra, we know the solution set is at least one-dimensional.

36 Uniqueness Uniqueness of such curve relies on Bézout s theorem (2 2 < 5). Namely, the given five points should be in the general position. Theorem Any five points in RP 2, no three of which are collinear, lie on exactly one conic. Theorem Any five points in RP 2, no four of which are collinear, lie on exactly one quadratic curve.

37 Uniqueness Uniqueness of such curve relies on Bézout s theorem (2 2 < 5). Namely, the given five points should be in the general position. Theorem Any five points in RP 2, no three of which are collinear, lie on exactly one conic. Theorem Any five points in RP 2, no four of which are collinear, lie on exactly one quadratic curve.

38 Uniqueness Uniqueness of such curve relies on Bézout s theorem (2 2 < 5). Namely, the given five points should be in the general position. Theorem Any five points in RP 2, no three of which are collinear, lie on exactly one conic. Theorem Any five points in RP 2, no four of which are collinear, lie on exactly one quadratic curve.

39 Cubic curve For a cubic curve, the same linear algebra reason suggests that any nine points can determine a cubic curve. What about the uniqueness? The issue is subtle as 9 = 3 3. Theorem (Cayley-Bacharach) Let P(x, y) = 0 and Q(x, y) = 0 be two cubic curves that intersect (over C) in precisely nine distinct points p 1,, p 9. Let R(x, y) be a cubic polynomial that vanishes on eight of these points (say p 1,, p 8 ). Then R is a linear combination of P and Q, in particular, R also vanishes on the ninth point p 9.

40 Cubic curve For a cubic curve, the same linear algebra reason suggests that any nine points can determine a cubic curve. What about the uniqueness? The issue is subtle as 9 = 3 3. Theorem (Cayley-Bacharach) Let P(x, y) = 0 and Q(x, y) = 0 be two cubic curves that intersect (over C) in precisely nine distinct points p 1,, p 9. Let R(x, y) be a cubic polynomial that vanishes on eight of these points (say p 1,, p 8 ). Then R is a linear combination of P and Q, in particular, R also vanishes on the ninth point p 9.

41 Pappus theorem Theorem (Pappus) Let l, l be two distinct lines, let A 1, A 2, A 3 be three distinct points on l not on l, and let B 1, B 2, B 3 be three distinct points on l not on l. Suppose that for ij = 12, 23, 31, the line A i B j and A j B i intersect at C ij. Then the three points C 12, C 23 and C 31 are collinear.

42 Pascal s theorem Theorem (Pascal) Let A 1, A 2, A 3, B 1, B 2, B 3 be distinct points on a conic σ. Suppose that for ij = 12, 23, 31, the lines A i B j and A j B i intersect at C ij. Then the points C 12, C 23 and C 31 are collinear.

43 References and further reading topics References: 1. Robert Bix, Conics and Cubics g Úng 2. Frances Kirwan, Complex Algebraic Curves E ê 3. Terence Tao, Pappus s theorem and elliptic curves, His blog: What s new? Figure out the multiplicity mentioned in this lecture. Why certain cubic curves are called Elliptic curves? Resultant