Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves Maunderings in enumerative geometry Gary Kennedy Ohio State University UC Santa Cruz, January 2015 Maunderings in enumerative geometry I \A rambling or pointless discourse" (Wiktionary) Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves \Maunderings" Maundering I \Rambling talk, drivel" (Oxford English Dictionary) Maunderings in enumerative geometry Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves \Maunderings" Maundering I \Rambling talk, drivel" (Oxford English Dictionary) I \A rambling or pointless discourse" (Wiktionary) Maunderings in enumerative geometry Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves Outline of Topics Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves Maunderings in enumerative geometry Franz Schubert (1797{1828) Hermann Schubert (1848{1911) Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves www.carus-verlag.com/ Maunderings in enumerative geometry Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves www.carus-verlag.com/ Franz Schubert (1797{1828) Hermann Schubert (1848{1911) Maunderings in enumerative geometry I For each term, its degree is the sum of its exponents. The maximum value is called the degree of the surface. Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves What's a surface? I A surface is the set of points in 3-dimensional space whose coordinates satisfy a single polynomial equation. Maunderings in enumerative geometry Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves What's a surface? I A surface is the set of points in 3-dimensional space whose coordinates satisfy a single polynomial equation. I For each term, its degree is the sum of its exponents. The maximum value is called the degree of the surface. Maunderings in enumerative geometry Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A sample problem I How many lines lie on the degree-2 surface x2 + y 2 − z2 − 1 = 0? http://www.math.umn.edu/ rogness/quadrics/hyp1sh.gif Maunderings in enumerative geometry I Example: Slice the surface using the plane x = 1 (the tangent plane at the point (1; 0; 0)). This gives the curve y 2 − z2 = 0, a pair of lines. I The same thing works at every point: if you slice using the tangent plane, you obtain a conic section. But it can't be a hyperbola or parabola or ellipse; it must be a degenerate type, which means that the equation of the curve can be factored into two linear equations. I Conclusion: this is a doubly ruled surface. Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A sample problem I Answer: infinitely many! Maunderings in enumerative geometry I The same thing works at every point: if you slice using the tangent plane, you obtain a conic section. But it can't be a hyperbola or parabola or ellipse; it must be a degenerate type, which means that the equation of the curve can be factored into two linear equations. I Conclusion: this is a doubly ruled surface. Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A sample problem I Answer: infinitely many! I Example: Slice the surface using the plane x = 1 (the tangent plane at the point (1; 0; 0)). This gives the curve y 2 − z2 = 0, a pair of lines. Maunderings in enumerative geometry I Conclusion: this is a doubly ruled surface. Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A sample problem I Answer: infinitely many! I Example: Slice the surface using the plane x = 1 (the tangent plane at the point (1; 0; 0)). This gives the curve y 2 − z2 = 0, a pair of lines. I The same thing works at every point: if you slice using the tangent plane, you obtain a conic section. But it can't be a hyperbola or parabola or ellipse; it must be a degenerate type, which means that the equation of the curve can be factored into two linear equations. Maunderings in enumerative geometry Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A sample problem I Answer: infinitely many! I Example: Slice the surface using the plane x = 1 (the tangent plane at the point (1; 0; 0)). This gives the curve y 2 − z2 = 0, a pair of lines. I The same thing works at every point: if you slice using the tangent plane, you obtain a conic section. But it can't be a hyperbola or parabola or ellipse; it must be a degenerate type, which means that the equation of the curve can be factored into two linear equations. I Conclusion: this is a doubly ruled surface. Maunderings in enumerative geometry I Huh? Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A caveat 2 2 2 I I make the same claim for the ellipsoid 4x + y + 4z − 4 = 0. www.math.hmc.edu/ gu Maunderings in enumerative geometry Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A caveat 2 2 2 I I make the same claim for the ellipsoid 4x + y + 4z − 4 = 0. www.math.hmc.edu/ gu I Huh? Maunderings in enumerative geometry 2 2 I 4x + y = (2x + iy)(2x − iy) I Thus the surface contains two lines through this point. I Conventional assumption of algebraic geometry: points are allowed to have complex numbers as coordinates, and equations are allowed to have complex numbers as coefficients. Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A caveat I At the topmost point (0; 0; 1) the tangent plane is z = 1, which meets the surface in the curve 4x2 + y 2 = 0. Maunderings in enumerative geometry I Thus the surface contains two lines through this point. I Conventional assumption of algebraic geometry: points are allowed to have complex numbers as coordinates, and equations are allowed to have complex numbers as coefficients. Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A caveat I At the topmost point (0; 0; 1) the tangent plane is z = 1, which meets the surface in the curve 4x2 + y 2 = 0. 2 2 I 4x + y = (2x + iy)(2x − iy) Maunderings in enumerative geometry I Conventional assumption of algebraic geometry: points are allowed to have complex numbers as coordinates, and equations are allowed to have complex numbers as coefficients. Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A caveat I At the topmost point (0; 0; 1) the tangent plane is z = 1, which meets the surface in the curve 4x2 + y 2 = 0. 2 2 I 4x + y = (2x + iy)(2x − iy) I Thus the surface contains two lines through this point. Maunderings in enumerative geometry Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a. the Monster tower) Tropical curves A caveat I At the topmost point (0; 0; 1) the tangent plane is z = 1, which meets the surface in the curve 4x2 + y 2 = 0. 2 2 I 4x + y = (2x + iy)(2x − iy) I Thus the surface contains two lines through this point. I Conventional assumption of algebraic geometry: points are allowed to have complex numbers as coordinates, and equations are allowed to have complex numbers as coefficients. Maunderings in enumerative geometry I A general surface of degree 4 or more contains no lines. I A general surface of degree 3 contains 27 lines. Counting lines on surfaces Counting rational curves in the plane Counting rational curves on the quintic hypersurface The Semple tower (a.k.a.