Intersection Theory I

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1 Jessica Sidman Mount Holyoke College Partial support from NSF grant DMS Clare Boothe Luce Program April 15, 2007

2 Systems of polynomial equations: varieties A homogeneous system of linear equations: x = 0 y z = 0 z = 0 A system of polynomial equations of higher degree: z xy = 0 y x 2 = 0

3 Our most basic goals: What is intersection theory supposed to do? H 1,..., H n hypersurfaces in an n-dimensional space H 1 H n = number of points in common to all H i X subvariety of dimension p Y subvariety of dimension n-p X Y = number of points in common

4 Applications of intersection theory Schubert conditions (How many lines in P 3 meet 4 given lines in general position?) Given 5 conics in P 2, how many conics are tangent to all 5?(3264) Bézout s theorem Understand the geometry of X P n : Does X contain lines? Can we enumerate them? How are the lines arranged? We ll discuss the last two.

5 How should H 1 H n behave? Example (H 1,..., H n A n, H i = V(f i ), deg f i = d i ) If H 1 = V(y x 2 ), H 2 = V(f 2 ), deg f 2 = 1, then H 1 H 2 should be 2. (a) y-1 (b) y-x-1 (c) y (d) y+1 (e) x We need to consider points of intersection 1. with multiplicity 2. in C 3. and if we also consider points at infinity...

6 Towards a definition of multiplicity Theorem (Bézout) If H 1,..., H n P n C, H i = V (f i ), deg(f i ) = d i, then H 1 H n = d 1 d n. F, G curves on a smooth surface X The multiplicity of F G at p, denoted mult p (F G), should be defined so that if curves meet transversely at p, then mult p (F G) = 1. intersection numbers are constant on families.

7 Looking locally at p x 0.5 y Definition The local ring of a point p in a variety X C n, denoted C[X] p, consists of all fractions f g where f, g C[X] and f g is defined in an open set containing p. Example (The local ring of the origin in C 2 ) C[C 2 ] = C[x, y], p = (0, 0) x (y x 2 1)(y+2) C[x, y] p is defined on the set with red curves deleted

8 Intersection multiplicity: smooth surfaces Definition (Multiplicity of an isolated point p F G) F, G curves on a smooth surface X C n, local equations f, g Define mult p (F G) = dim C[X] p / f, g, and F G = p F G mult p(f G). Example (A transverse intersection in C 2 ) x = 0, y x 2 = 0, p = (0, 0) If r s C[x, y] p, then there are r ij, s ij C, s 00 0 so that r(x, y) = r 00 + r 10 x + r 01 y + r 20 x 2 + r 11 xy +, s(x, y) = s 00 + s 10 x + s 01 y + s 20 x 2 + s 11 xy +. C[x, y] p / x, y x 2 = C since x = 0 and y = x 2 = 0 r s r 00 s 00.

9 Intersections with multiplicity > 1. Example y = 0, y x 2 = 0, p = (0, 0) C[x, y] p / y, y x 2 is spanned by {1, x}. Since y = 0 and x 2 = y = 0, r s C[x, y] p, maps to r 00 + r 10 x s 00 + s 10 x = r 00 + r 10 x s 00 + s 10 x s00 s 10 x s 00 s 10 x = r 00s 00 + (r 10 s 00 r 00 s 10 )x s 2 00 Therefore, mult p (F G) = 2.

10 The degree of a curve in P 2 y = x 3 x crosses the x-axis at 3 points x y C = V(yz 2 x 3 xz 2 ) P 2 For any line L P 2, 3 = C L. deg C = 3

11 The degree of a variety X P n variety, dim X = p H general plane, dim H = n p The degree of X is X H. Example (The degree of a hypersurface I) X = V(f ) P 3, f = w 0 w 3 w 1 w 2 L = {[s : s : s : t]} f ([s : s : s : t]) = st s 2 = s(t s). X L : {[0 : 0 : 0 : 1], [1 : 1 : 1 : 1]} deg X = X L = 2

12 Projective embeddings: Veronese varieties Definition The d-uple Veronese embedding ν d : P n P N is given by x = [x 0 : : x n ] [F 0 (x) : : F N (x)], where N = ( ) d+n n 1 and F0,..., F N is a basis for the homogeneous forms of degree d on P n. Example (Veronese embeddings of P 1 : rational normal curves) ν 2 ([x : y]) = [x 2 : xy : y 2 ] f = w 0 w 2 w 2 1 f ([x 2 : xy : y 2 ]) = (x 2 )(y 2 ) (xy) 2 = 0

13 The degree of ν d (P n ) P N Example (deg ν 3 (P 1 ) P 3 ) ν 3 ([x : y]) = [x 3 : x 2 y : xy 2 : y 3 ] L = w 0 w 2 L(x 3, x 2 y, xy 2, y 3 ) = x 3 xy 2 has three solutions To compute deg ν d (P n ) : H = H 1 H n where H i is defined by a linear form L i. ν d (P n ) H = ν d (P n ) H 1 H n L i (x0 d, x d 1 0 x 1,..., xn d ) is a homogeneous form of degree d Bézout s theorem hypersurfaces in P n defined by pulling back equations L i intersect in d n points. Therefore, deg ν d (P n ) = d n.

14 Maps with basepoints I Example (Conics through a point) Let p = [1 : 0 : 0] in P 2 and let W p be the vector space of homogeneous forms {F C[x, y, z] deg F = 2, F(p) = 0}. W p = span{xy, xz, y 2, yz, z 2 }. Define φ : P 2 \{p} P 4 by φ([x : y : z]) = [xy : xz : y 2 : yz : z 2 ]. We say that φ has a basepoint at p since it is not defined at p. What is deg Image φ?

15 Maps with basepoints II Example (Conics through 2 points) Fix p = [1 : 0 : 0], q = [0 : 1 : 0]. Define ψ : P 2 \{p, q} P 3 by ψ([x : y : z]) = [xy : xz : yz : z 2 ]. Example (Cubics through 6 points) Fix p 1,..., p 6 P 2. Define γ : P 2 \{p 1,..., p 6 } P 3 using a basis of cubics passing through the 6 points. Question What can we say about Image φ, Image ψ, Image γ? What are their degrees? How do their geometries differ?