Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007
Outline Introduction Curves Surfaces Curves on surfaces
Introduction Algebraic geometry = study of common solutions of a given set of polynomial equations. The equations 3x + 5y = 1 and 2x 9y = 13 have the common solution x = 2 and y = 1
What is so fascinating about solving equations? The interplay between algebraic manipulations and geometric interpretation. The above solution can be found by (linear) algebra, or by geometry (the intersection of two lines). ALGEBRA + GEOMETRY = TRUE
What is a curve? Intuitively, a curve is a geometric object of dimension one: a moving particle traces a curve. For example, the leminiscate is given as the set of points (x, y) in the plane such that x = Is this an algebraic curve? sin t 1+cos 2 t and y = sin t cos t 1+cos 2 t
Yes: as a point set, it is equal to the plane curve given by the equation (x 2 + y 2 ) 2 = x 2 y 2 and it has the rational parameterization x = u + u3 u u3 and y = 1 + u4 1 + u 4 The parameterized curve x = t, y = sin t is not algebraic: it intersects the x-axis in infinitely many points.
Projective curves An algebraic variety V = V(f 1,..., f r ) P n (k) is the set of common zeros of finitely many homogeneous polynomials. A rational function on V is the restriction to V of a rational function f g on Pn (k), where f and g are polynomials of the same degree. The function field K = K(V ) is the set of rational functions on V. The dimension of V is equal to the transcendence degree of K over k. V is a curve if tr deg k (K) = 1. Nonsingular projective curves with the same function field are isomorphic. Any algebraic curve has a unique nonsingular model.
The arithmetic genus Recall that the Hilbert polynomial P (t) of a variety V P n (k) satisfies (for m big enough) P (m) = ( ) n+m n dimk I(V ) m. The arithmetic genus p a of V is defined to be p a = ( 1) dim V (P (0) 1) If V is a curve, P (t) = dt + 1 p a, where d is the degree of V. Example. Let V P 3 be the twisted cubic, given by u (1 : u : u 2 : u 3 ). Then the ideal I(V ) is generated by the 2-minors of the matrix [ ] x0 x 1 x 2 x 1 x 2 x 3 We find P (t) = 3t + 1 and p a = 0.
Plane curves A plane curve: V = V(f) P 2 (k), where f is a homogeneous polynomial. The degree d of V is the degree of f. P (t) = ( ) ( t+2 2 t d+2 ) ( 2 = dt + 1 d 1 ) 2 The arithmetic genus of a plane curve of degree d is p a = (d 1)(d 2)/2 Here are real (k = R) curves of arithmetic genus 1 and 0:
Topological genus A complex (k = C) algebraic curve can be viewed as a two-dimensional real manifold, a Riemann surface. Topological genus: g=# holes in the Riemann surface Euler number: e = # vertices # edges +# faces in a triangulation Euler Poincaré formula: e = b 0 b 1 + b 2 = 1 2g + 1 = 2 2g (The b i are the Betti numbers.) The first curve has topological genus 1, the second curve has topological genus 0.
Hirzebruch Riemann Roch: Topological genus = Arithmetic genus Proof. Project V P n to a plane curve V P 2 with δ double points. Algebra. Compare the arithmetic genus of V with that of V : p a (V ) = p a (V ) δ = (d 1)(d 2) 2 δ Geometry. Compute the number d of tangents to V through a given point P : d = d(d 1) 2δ
Topology. Project V from P to get a map V P 1. The map is a topological d-fold cover, with d ramification points. Compare the topological genus of V with that of P 1 : Triangulate the sphere P 1 and lift the triangulation to V : e(v ) = (d#{vertices of P 1 } d ) d#{edges of P 1 }+d#{faces of P 1 } e(v ) = d e(p 1 ) d 2 2g = d(2 2 0) d = 2d d
Algebra: p a = 1 (d 1)(d 2) δ 2 Geometry: Topology: d = d(d 1) 2δ 2 2g = 2d d Algebra + Geometry + Topology 1 p a = 1 g
Classification of curves Have noted that all nonsingular projective curves with the same function field are isomorphic and that any curve has a unique nonsingular model. Classification problem: describe the moduli space of all curves of genus g. A curve of genus 0 is called rational. The nonsingular model of any rational curve is P 1. All tori are topologically the same, but: There is a one-dimensional family of algebraic curves of genus 1. Indeed, all Riemann surfaces of the same genus are topologically the same, but: The moduli space of algebraic curves of genus g has dimension 3g 3, for g 2.
Classification of curves in a fixed space The Hilbert scheme parameterizes the set of curves of given degree and given arithmetic genus, in a given projective space. The Hilbert scheme of degree d curves in P 2 is the projective space P d(d+3)/2. Indeed, a curve of degree d is given by a homogenous polynomial in three variables, of degree d. The space of such polynomials has dimension ( ) d + 2 d(d + 3) = + 1. 2 2 The Hilbert scheme of degree 3 curves of genus 0 in P 3 is the union of a nonsingular 12-dimensional and a nonsingular 15-dimensional algebraic variety, intersecting along a nonsingular 11-dimensional variety.
Surfaces A surface is a variety V P n (k) of dimension 2. Dimension 2 means that the transcendence degree over k of the function field K(V ) is 2. The arithmetic genus: p a = P (0) 1 Topologically, a surface is a four-dimensional real variety. The Euler number e = ( 1) i b i is a topological invariant. The genus p of a canonical curve on V is another topological invariant.
Noether s formula 1 + p a = 1 (p 1 + e) 12 Proof. Project V to a surface V P 3 with generic singularities (a double curve and finitely many triple points and pinch points).
Algebra. Compute the arithmetic genus of V in terms of the arithmetic genera of V, the double curve, and the inverse image of the double curve ( conductor square ). Geometry. Compute the class of V : the number of tangent planes to V through a given line L. Compute the rank of V : the number of tangent planes to V at points in H V through a given point P. Topology. Compute the ramification curve of the projection V P 2 in terms of topological invariants of V and P 2. Algebra + Geometry + Topology = OK
Classification of surfaces Given a function field K of transcendence degree 2 over k, there are infinitely many surfaces V such that K(V ) = K. The reason for this is that one can blow up a point on a surface: replace the point by the set of all the tangent directions through that point. This set is a P 1, and is called an exceptional curve. A surface which contains no exceptional curves is called minimal. Almost all surfaces have a unique minimal model. Enriques Kodaira classification: κ = : rational surfaces and ruled surfaces over curves of genus > 0 κ = 0 : Enriques surfaces, K3 sufaces, and elliptic surfaces κ = 1 : elliptic fibrations over a curve of genus 2 κ = 2 : surfaces of general type
Curves on surfaces Try to describe a variety by describing its subvarieties. Given a surface V, what kind of curves does it contain? For example, a surface V P n that contains infinitely many lines is a ruled surface (κ = ). Can any surface V be covered by rational curves? Are there surfaces containing no rational curves? If there are only finitely many rational curves, can one count them?
Rational curves on P 2 A curve in P 2 of degree d with only δ double points as singularities is rational (has genus 0) if and only if δ = (d 1)(d 2)/2. The set of plane curves of degree d is P d(d+3)/2. The curves with (d 1)(d 2)/2 double points is a subvariety of dimension d(d + 3)/2 (d 1)(d 2)/2 = 3d 1. Let N d denote the number of plane rational curves of degree d passing through 3d 1 given points. N 1 = 1, N 2 = 1, N 3 = 12, N 4 = 620,...
Kontsevich s recursion formula: N d = d 1 +d 2 =d N d1 N d2 ( d 2 1 d 2 2 ( ) ( ) 3d 4 3d 4 ) d 3 3d 1 2 1d 2 3d 1 1 If we set n d := n d = d 1 +d 2 =d N d (3d 1)!, then n d1 n d2 d 1 d 2 ((3d 1 2)(3d 2 2)(d + 2) + 8(d 1)) 6(3d 1)(3d 2)(3d 3) Kontsevich s proof used a degeneration argument: mapping the one-dimensional family of curves through 3d 2 points to P 1 using the cross-ratio, and counting degrees above 0 and 1.
Quadric surfaces A quadric surface V P 3 is isomorphic to P 1 P 1, embedded via the Segre map: (s : t) (u : v) (su : sv : tu : tv) There are two families of lines on V. Through each point of V pass two lines, one from each family.
Lines on a cubic surface A cubic surface V P 3 contains 27 lines! Every cubic surface is isomorphic to P 2 blown up in 6 points. The 6 exceptional curves are six of the 27 lines. The transforms of the 15 lines through two of the six points are also among the 27 lines. The last 6 lines are the transforms of the conics through 5 of the 6 points. 6 + ( ) 6 + 2 ( ) 6 = 6 + 15 + 6 = 27 5
Combinatorial sidestep: partitions Let p denote the partition function: p(n) is the number of ways of writing n = n 1 +... + n k, where n 1 n k 1 The generating function for p, ϕ(q) := n 0 p(n)q n, is equal to the formal power series Π m 1 (1 q m ) 1 = 1 + q + 2q 2 + 3q 3 + 5q 4 +....
Rational curves on a quartic surface Let V P 3 be a surface of degree 4. For each integer r, let C V be a curve such that C 2 /2 = r 1, and set N r := #{ rational curves with r double points, linearly equivalent to C} The generating function for the integers N r is f(q) = r N r q r Surprise (Yau Zaslow): f(q) = Π(1 q m ) 24 = ϕ(q) 24 where ϕ is the generating function of the partition function.
Proof. (Bryan Leung) Pass to symplectic geometry! Degenerate V to a fibration V P 1 where a general fiber has genus 1, with 24 fibers that have one double point and hence are rational, and C to C = S + rf, where S = s(p 1 ) is a section and F is a fiber. Want to count curves that degenerate to S + 24 i=1 a if i, where F 1,..., F 24 are the fibers with one double point, and where 24 i=1 a i = r. There are p(a i ) ways to degenerate to a i F i, hence p(a 1 ) p(a 24 ) ways to degenerate to S + 24 i=1 a if i. Therefore N r is the coefficient of q r in 24th power of the generating function ϕ(q).
String theory and enumerative geometry A quartic surface in P 3 is an example of a Calabi Yau variety. Calabi Yau varieties appear naturally in the superstring model of the universe. Rational curves on a (three-dimensional) Calabi Yau variety are interpreted as instantons. Physicists predicted the generating function for instantons on a three-dimensional variety in P 4 of degree 5. This lead to a boom for classical enumerative geometry, using new tools coming from physics mirror symmetry, quantum cohomology, and Gromov Witten theory. The interaction between algebraic geometers, symplectic geometers, and theoretical physicists has been intense and surprising, with mutual benefits.
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