* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Algebraic Geometry I
Survey
Document related concepts
Duality (mathematics) wikipedia , lookup
Dessin d'enfant wikipedia , lookup
Algebraic geometry wikipedia , lookup
Euclidean geometry wikipedia , lookup
Cartan connection wikipedia , lookup
Metric tensor wikipedia , lookup
Rational trigonometry wikipedia , lookup
Riemannian connection on a surface wikipedia , lookup
System of polynomial equations wikipedia , lookup
Analytic geometry wikipedia , lookup
Quadratic form wikipedia , lookup
Algebraic variety wikipedia , lookup
Lie sphere geometry wikipedia , lookup
Duality (projective geometry) wikipedia , lookup
Transcript
Algebraic Geometry I - Problem Set 8 Write up solutions to three of the problems (write as legibly and clearly as you can, preferably in LaTeX). 1. (Intersection Multiplicities.) Let C = V (f ) and D = V (g) be two distinct curves in A2 . Recall that the multiplicity of intersection mp (C, D) of C and D at p is defined as the dimension of the k-vector space OA2 ,p / < f, g >. Prove that: (a) mp (C, D) ≥ mp (C) · mp (D). (b) If p ∈ C, for all but finitely many lines L through p, mp (C, L) = mp (C). (c) If C = C1 ∪ . . . ∪ Cr and D = D1 ∪ . . . ∪ Ds are decompositions into irreducible components, then X mp (C, D) = mp (Ci , Dj ) i,j 2. Consider the space P5 parametrizing all conics in P2 as follows: to any [a, b, c, d, e, f ] ∈ P5 we associate the symmetric quadratic form whose matrix of coefficients in given by: a b c b d e c e f In other words, it corresponds to the conic with equation: ax20 + 2bx0 x1 + 2cx0 x2 + dx21 + 2ex1 x2 + f x22 = 0 (Assume that the characteristic is not 2.) Let Σ, respectively Γ, be the locus parametrizing conics for which the corresponding quadratic polynomial is reducible, respectively the square of a linear form (i.e., this is the locus of matrices of rank ≤ 2, resp. rank ≤ 1). (a) Prove that Γ is the image of the Veronese map v2 : P̌2 → P5 , where P̌2 denotes the dual P2 , i.e., the space G(1, 2) of lines in P2 (a line ax0 + bx1 + cx2 = 0 corresponds in P̌2 to the point [a, b, c]). (b) Prove that Σ is a cubic hypersurface. (c) Prove that Σ is the secant variety to Γ, i.e., the closure of the union of all the lines in P5 that intersect Γ in at least 2 points. 3. Let C = V+ (F ) ⊂ P2 be a smooth curve (F is a homogeneous polynomial in k[X, Y, Z]). Consider the morphism: ∂f ∂f ∂f φC : C → P2 , p 7→ (p), (p), (p) . ∂x0 ∂x1 ∂x2 The image φ(C) is called the dual curve of C. (a) Find a geometric description of φ. What does it mean geometrically that φ(p) = φ(q) for two distinct points p, q ∈ C? (b) If C is a conic, prove that its dual φ(C) is also a conic. (c) For any five lines in P2 in general position (what does this mean?) show that there is a unique conic in P2 that is tangent to these five lines. 1 2 4. Assume char k 6= 2, 3. Prove that a smooth plane cubic C has nine distinct inflection points. Equivalently prove that mp (C, Hess(C)) = 1 for every inflection point p of C. (Hint: you may assume that the inflection point p is [0, 0, 1] and the tangent line to C at p is y = 0. Then prove that if f = f (x, y) is the local equation of C then in the local ring OA2 ,p , we have < f, H(x, y, 1) >=< x, y >. Note that H(x, y, 1) = yp(x, y) + H(x, 0, 1) and H(x, 0, 1) = xv, where v is a unit in OA2 ,p . Alternatively, you may use the Weierstrass normal form of the cubic to prove this.) 5. (The j-invariant.) Prove that j(λ) = j(µ) if and only if Eλ ∼ = Eµ . Hint: prove that j(λ) = j(µ) if and only if 1 1 λ λ−1 µ ∈ {λ, , 1 − λ, , , }. λ 1−λ λ−1 λ 6. Let C be a smooth cubic and let O ∈ C. Consider the group structure (C, ⊕) that has O as the identity. Prove that: (i) The inflection points of C correspond to the torsion points of order 3. (ii) If P, Q are inflection points and R is the third point of intersection of the line P Q with C, then R is an inflection point. (iii) Prove that if O ∈ C is an inflection point, then the nine inflection points form a subgroup of C that is isomorphic to (Z/3Z)2 .