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. the Monster tower) Tropical curves Counting lines on surfaces

I A general surface of degree 2 contains infinitely many lines.

Maunderings in enumerative geometry 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. the Monster tower) Tropical curves Counting lines on surfaces

I A general surface of degree 2 contains infinitely many lines.

I A general surface of degree 4 or more contains no lines.

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 Counting lines on surfaces

I A general surface of degree 2 contains infinitely many lines.

I A general surface of degree 4 or more contains no lines.

I A general surface of degree 3 contains 27 lines.

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 The 27 lines on a cubic

www.cubics.algebraicsurface.net

3 3 3 I Example: On the Fermat cubic x + y + z − 1 = 0, here are 9 of the 27 lines:

x = α, z = −βy

where α and β are cube roots of 1.

Maunderings in enumerative geometry I The maximum degree of p, q, and r is called the degree of the curve. (Equivalently, it’s the degree of the equation in x and y obtained by eliminating t.)

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 rational curve?

I A rational plane curve is a plane curve for which there is a parametrization by rational functions

p(t) q(t) x = , y = r(t) r(t)

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 rational curve?

I A rational plane curve is a plane curve for which there is a parametrization by rational functions

p(t) q(t) x = , y = r(t) r(t)

I The maximum degree of p, q, and r is called the degree of the curve. (Equivalently, it’s the degree of the equation in x and y obtained by eliminating t.)

Maunderings in enumerative geometry 2 3 2 I Intrinsic equation: y = x + x

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 rational cubic

I Example of a rational curve of degree 3

t2 − 1 t3 − t x = , y = 1 1

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 rational cubic

I Example of a rational curve of degree 3

t2 − 1 t3 − t x = , y = 1 1

2 3 2 I Intrinsic equation: y = x + x

Maunderings in enumerative geometry I The surface will have a number of holes. This is called its genus.

I Rational curves are those of genus 0.

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 Rational = genus 0

I If you look at all the complex-valued points of a plane curve it’s really a surface. (Basic idea: one complex dimension is the same as two real dimensions.)

Maunderings in enumerative geometry I Rational curves are those of genus 0.

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 Rational = genus 0

I If you look at all the complex-valued points of a plane curve it’s really a surface. (Basic idea: one complex dimension is the same as two real dimensions.)

I The surface will have a number of holes. This is called its genus.

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 Rational = genus 0

I If you look at all the complex-valued points of a plane curve it’s really a surface. (Basic idea: one complex dimension is the same as two real dimensions.)

I The surface will have a number of holes. This is called its genus.

I Rational curves are those of genus 0.

Maunderings in enumerative geometry I Thus a natural question of enumerative geometry is this: how many rational plane curves of degree d pass through 3d − 1 general points?

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 Counting rational plane curves

I When specifying a rational curve, you have a lot of freedom in specifying the rational functions. One can show there are 3d − 1 degrees of freedom, where d is the degree.

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 Counting rational plane curves

I When specifying a rational curve, you have a lot of freedom in specifying the rational functions. One can show there are 3d − 1 degrees of freedom, where d is the degree.

I Thus a natural question of enumerative geometry is this: how many rational plane curves of degree d pass through 3d − 1 general points?

Maunderings in enumerative geometry I Kontsevich’s insight: Consider all the problems simultaneously. He found a simple recursion for computing the answers.

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 Counting rational plane curves

d 3d − 1 N(d) 1 2 1 2 5 1 3 8 12 4 11 620 Zeuthen – 19th century 5 14 87304 Ran & Vainsencher – early 1990s 6 17 26312976 Kontsevich – 1994 7 20 14616808192 Kontsevich – 1994

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 Counting rational plane curves

d 3d − 1 N(d) 1 2 1 2 5 1 3 8 12 4 11 620 Zeuthen – 19th century 5 14 87304 Ran & Vainsencher – early 1990s 6 17 26312976 Kontsevich – 1994 7 20 14616808192 Kontsevich – 1994

I Kontsevich’s insight: Consider all the problems simultaneously. He found a simple recursion for computing the answers.

Maunderings in enumerative geometry I Example: The number of plane rational cubics through 2 specified points and tangent to 6 specified lines is 756.

I Joined by Colley, we looked at similar questions, where we demanded high-order tangency to specified lines or curves.

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 My work with Ernstr¨omand Colley

I Ernstr¨omand I asked: what if some of the 3d − 1 “point conditions” are replaced by “line conditions” (meaning that the curve must be tangent to specified lines)?

Maunderings in enumerative geometry I Joined by Colley, we looked at similar questions, where we demanded high-order tangency to specified lines or curves.

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 My work with Ernstr¨omand Colley

I Ernstr¨omand I asked: what if some of the 3d − 1 “point conditions” are replaced by “line conditions” (meaning that the curve must be tangent to specified lines)?

I Example: The number of plane rational cubics through 2 specified points and tangent to 6 specified lines is 756.

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 My work with Ernstr¨omand Colley

I Ernstr¨omand I asked: what if some of the 3d − 1 “point conditions” are replaced by “line conditions” (meaning that the curve must be tangent to specified lines)?

I Example: The number of plane rational cubics through 2 specified points and tangent to 6 specified lines is 756.

I Joined by Colley, we looked at similar questions, where we demanded high-order tangency to specified lines or curves.

Maunderings in enumerative geometry I One possible answer comes from string theory in theoretical physics. For complicated reasons, they want to know how to count rational curves on “Calabi-Yau hypersurfaces.”

I The simplest example is the hypersurface of degree 5 in 4-dimensional space (called the quintic 3-fold).

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 One motivation

I Why study such questions?

Maunderings in enumerative geometry I The simplest example is the hypersurface of degree 5 in 4-dimensional space (called the quintic 3-fold).

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 One motivation

I Why study such questions?

I One possible answer comes from string theory in theoretical physics. For complicated reasons, they want to know how to count rational curves on “Calabi-Yau hypersurfaces.”

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 One motivation

I Why study such questions?

I One possible answer comes from string theory in theoretical physics. For complicated reasons, they want to know how to count rational curves on “Calabi-Yau hypersurfaces.”

I The simplest example is the hypersurface of degree 5 in 4-dimensional space (called the quintic 3-fold).

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.

I Example of a quintic 3-fold (hypersurface of degree 5 in 4-dimensional space): w 5 + x5 + y 5 + z5 − wxyz = 0.

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 hypersurface?

I A hypersurface is the set of points in n-dimensional space whose coordinates satisfy a single polynomial equation.

Maunderings in enumerative geometry I Example of a quintic 3-fold (hypersurface of degree 5 in 4-dimensional space): w 5 + x5 + y 5 + z5 − wxyz = 0.

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 hypersurface?

I A hypersurface is the set of points in n-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 What’s a hypersurface?

I A hypersurface is the set of points in n-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.

I Example of a quintic 3-fold (hypersurface of degree 5 in 4-dimensional space): w 5 + x5 + y 5 + z5 − wxyz = 0.

Maunderings in enumerative geometry I Clemens’s conjecture: This expectation is fulfilled.

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 so special about a quintic 3-fold?

I This is the only combination of dimension and degree for which one expects that the number of rational curves of degree d is finite, for every d. This expectation is based on a naive counting of “conditions.”

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 so special about a quintic 3-fold?

I This is the only combination of dimension and degree for which one expects that the number of rational curves of degree d is finite, for every d. This expectation is based on a naive counting of “conditions.”

I Clemens’s conjecture: This expectation is fulfilled.

Maunderings in enumerative geometry I It gave n1 = 2875, n2 = 609250 (both known to be correct), n3 = 317206375 (later confirmed). I Current mathematical consensus: The string theorists are counting something slightly different (“instanton numbers”). Their arguments mix rigorous calculations with physical intuition and sophisticated numerology. Rigorous statements and proofs have now been given up to degree 10.

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 Counting rational curves on a quintic 3-fold

I In 1991, four string theorists (Candela, de la Ossa, Green, Parkes) announced a recursion.

Maunderings in enumerative geometry I Current mathematical consensus: The string theorists are counting something slightly different (“instanton numbers”). Their arguments mix rigorous calculations with physical intuition and sophisticated numerology. Rigorous statements and proofs have now been given up to degree 10.

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 Counting rational curves on a quintic 3-fold

I In 1991, four string theorists (Candela, de la Ossa, Green, Parkes) announced a recursion.

I It gave n1 = 2875, n2 = 609250 (both known to be correct), n3 = 317206375 (later confirmed).

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 Counting rational curves on a quintic 3-fold

I In 1991, four string theorists (Candela, de la Ossa, Green, Parkes) announced a recursion.

I It gave n1 = 2875, n2 = 609250 (both known to be correct), n3 = 317206375 (later confirmed). I Current mathematical consensus: The string theorists are counting something slightly different (“instanton numbers”). Their arguments mix rigorous calculations with physical intuition and sophisticated numerology. Rigorous statements and proofs have now been given up to degree 10.

Maunderings in enumerative geometry I We can also do this for certain singular curves, by taking limits.

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 Curvilinear data

I Suppose C is a smooth plane curve passing through the origin, and that its tangent line there isn’t vertical. Then we can associate to it a sequence of curvilinear data, the values at the origin of dy d2y dny , ,..., . dx dx2 dxn

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 Curvilinear data

I Suppose C is a smooth plane curve passing through the origin, and that its tangent line there isn’t vertical. Then we can associate to it a sequence of curvilinear data, the values at the origin of dy d2y dny , ,..., . dx dx2 dxn I We can also do this for certain singular curves, by taking limits.

Maunderings in enumerative geometry 2 4 5 I Parametrize using x = t and y = t + t . I Calculate dy 5 y 0 = = 2t2 + t3 → 0 (as t → 0) dx 2 dy 0 15 y 00 = = 2 + t → 2 dx 4

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 Curvilinear data of a singular curve

2 2 5 I Example: (y − x ) = x

Maunderings in enumerative geometry I Calculate dy 5 y 0 = = 2t2 + t3 → 0 (as t → 0) dx 2 dy 0 15 y 00 = = 2 + t → 2 dx 4

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 Curvilinear data of a singular curve

2 2 5 I Example: (y − x ) = x

2 4 5 I Parametrize using x = t and y = t + t .

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 Curvilinear data of a singular curve

2 2 5 I Example: (y − x ) = x

2 4 5 I Parametrize using x = t and y = t + t . I Calculate dy 5 y 0 = = 2t2 + t3 → 0 (as t → 0) dx 2 dy 0 15 y 00 = = 2 + t → 2 dx 4

Maunderings in enumerative geometry I But at the next step of the calculation,

dy 00 15 = → ∞. dx 8t

I What does this mean? How do we deal with it?

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 Curvilinear data of a singular curve

I This calculation shows that the curve has the same curvilinear data up to second order as the parabola y = x2. dy 5 y 0 = = 2t2 + t3 → 0 dx 2 dy 0 15 y 00 = = 2 + t → 2 dx 4

Maunderings in enumerative geometry I What does this mean? How do we deal with it?

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 Curvilinear data of a singular curve

I This calculation shows that the curve has the same curvilinear data up to second order as the parabola y = x2. dy 5 y 0 = = 2t2 + t3 → 0 dx 2 dy 0 15 y 00 = = 2 + t → 2 dx 4

I But at the next step of the calculation,

dy 00 15 = → ∞. dx 8t

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 Curvilinear data of a singular curve

I This calculation shows that the curve has the same curvilinear data up to second order as the parabola y = x2. dy 5 y 0 = = 2t2 + t3 → 0 dx 2 dy 0 15 y 00 = = 2 + t → 2 dx 4

I But at the next step of the calculation,

dy 00 15 = → ∞. dx 8t

I What does this mean? How do we deal with it?

Maunderings in enumerative geometry I But what do we do in general? Once we’ve gotten infinite data at some order, can we then get finite data, and if so what does that mean?

I I.e., what is the appropriate way to compactify the space of curvilinear data?

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 Curvilinear data of a singular curve

I Here the “obvious fix” is to turn things upside down: dx 8t = → 0. dy 00 15

Maunderings in enumerative geometry I I.e., what is the appropriate way to compactify the space of curvilinear data?

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 Curvilinear data of a singular curve

I Here the “obvious fix” is to turn things upside down: dx 8t = → 0. dy 00 15

I But what do we do in general? Once we’ve gotten infinite data at some order, can we then get finite data, and if so what does that mean?

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 Curvilinear data of a singular curve

I Here the “obvious fix” is to turn things upside down: dx 8t = → 0. dy 00 15

I But what do we do in general? Once we’ve gotten infinite data at some order, can we then get finite data, and if so what does that mean?

I I.e., what is the appropriate way to compactify the space of curvilinear data?

Maunderings in enumerative geometry I It leads to the Semple tower S(n) → S(n − 1) → ... S(2) → S(1) → S(0) = the plane

I Each fiber is a projective line, one of whose points represents “infinite data.”

I Can be used to solve problems of contact 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 The Semple tower

I There is a beautifully simple way to compactify the spaces of curvilinear data, originally due to Semple.

Maunderings in enumerative geometry I Each fiber is a projective line, one of whose points represents “infinite data.”

I Can be used to solve problems of contact 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 The Semple tower

I There is a beautifully simple way to compactify the spaces of curvilinear data, originally due to Semple.

I It leads to the Semple tower S(n) → S(n − 1) → ... S(2) → S(1) → S(0) = the plane

Maunderings in enumerative geometry I Can be used to solve problems of contact 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 The Semple tower

I There is a beautifully simple way to compactify the spaces of curvilinear data, originally due to Semple.

I It leads to the Semple tower S(n) → S(n − 1) → ... S(2) → S(1) → S(0) = the plane

I Each fiber is a projective line, one of whose points represents “infinite data.”

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 The Semple tower

I There is a beautifully simple way to compactify the spaces of curvilinear data, originally due to Semple.

I It leads to the Semple tower S(n) → S(n − 1) → ... S(2) → S(1) → S(0) = the plane

I Each fiber is a projective line, one of whose points represents “infinite data.”

I Can be used to solve problems of contact in enumerative geometry.

Maunderings in enumerative geometry I By counting parameters, we see that the curvilinear data of points on conics can’t account for all possible points on S(5). Thus there is a sextactic locus of such points. I Intersecting this locus with the curvilinear data points of C (and using the apparatus of intersection theory in algebraic geometry), one can calculate how many points on C have “unexpectedly good agreement” with some conic. I For a general smooth curve of degree d > 3, there are 3d(4d − 9) such points.

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 Sextactic points

I Here’s an example. Consider a point on a smooth plane curve C. Can we find a conic (degree 2 curve) having the same curvilinear data up to order 3? . . . up to order 4? . . . up to order 5?

Maunderings in enumerative geometry I Intersecting this locus with the curvilinear data points of C (and using the apparatus of intersection theory in algebraic geometry), one can calculate how many points on C have “unexpectedly good agreement” with some conic. I For a general smooth curve of degree d > 3, there are 3d(4d − 9) such points.

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 Sextactic points

I Here’s an example. Consider a point on a smooth plane curve C. Can we find a conic (degree 2 curve) having the same curvilinear data up to order 3? . . . up to order 4? . . . up to order 5? I By counting parameters, we see that the curvilinear data of points on conics can’t account for all possible points on S(5). Thus there is a sextactic locus of such points.

Maunderings in enumerative geometry I For a general smooth curve of degree d > 3, there are 3d(4d − 9) such points.

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 Sextactic points

I Here’s an example. Consider a point on a smooth plane curve C. Can we find a conic (degree 2 curve) having the same curvilinear data up to order 3? . . . up to order 4? . . . up to order 5? I By counting parameters, we see that the curvilinear data of points on conics can’t account for all possible points on S(5). Thus there is a sextactic locus of such points. I Intersecting this locus with the curvilinear data points of C (and using the apparatus of intersection theory in algebraic geometry), one can calculate how many points on C have “unexpectedly good agreement” with some conic.

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 Sextactic points

I Here’s an example. Consider a point on a smooth plane curve C. Can we find a conic (degree 2 curve) having the same curvilinear data up to order 3? . . . up to order 4? . . . up to order 5? I By counting parameters, we see that the curvilinear data of points on conics can’t account for all possible points on S(5). Thus there is a sextactic locus of such points. I Intersecting this locus with the curvilinear data points of C (and using the apparatus of intersection theory in algebraic geometry), one can calculate how many points on C have “unexpectedly good agreement” with some conic. I For a general smooth curve of degree d > 3, there are 3d(4d − 9) such points.

Maunderings in enumerative geometry I The fact that the constructions are the same was first realized by Alex Castro.

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 Rediscovery

I Differential geometers studying manifolds with distributions — Montgomery, Zhitomirskii, et al. — independently discovered this construction, dubbing it the Monster tower.

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 Rediscovery

I Differential geometers studying manifolds with distributions — Montgomery, Zhitomirskii, et al. — independently discovered this construction, dubbing it the Monster tower.

I The fact that the constructions are the same was first realized by Alex Castro.

Maunderings in enumerative geometry I We begin to analyze the tower by a coarse classification: to each point is attached a code word. In the algebro-geometric code invented with Colley, we use the symbols 0, –, *, and ∞. The differential geometers invented a code with symbols R (“regular”), V (“vertical”), and T (“tangent”). There is an elementary translation between the codes.

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 How our projects agree

I We do the same constructions.

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 How our projects agree

I We do the same constructions.

I We begin to analyze the tower by a coarse classification: to each point is attached a code word. In the algebro-geometric code invented with Colley, we use the symbols 0, –, *, and ∞. The differential geometers invented a code with symbols R (“regular”), V (“vertical”), and T (“tangent”). There is an elementary translation between the codes.

Maunderings in enumerative geometry I For enumerative geometry, one acts by the 8-dimensional group PGL(3). The problems get increasingly more intricate as one acquires more and more orbits. Eventually there are infinitely many orbits.

I In the differential-geometric setting, it’s not obvious whether the orbit space is finite (at each fixed level) or if there are moduli. In fact there are moduli, but when and where do they enter?

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 How our projects differ

I The differential geometers “can’t see” (or don’t care about) things like conics or the sextactic locus. That’s because they want to act on the space by the full group of local diffeomorphisms (an infinite-dimensional group).

Maunderings in enumerative geometry I In the differential-geometric setting, it’s not obvious whether the orbit space is finite (at each fixed level) or if there are moduli. In fact there are moduli, but when and where do they enter?

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 How our projects differ

I The differential geometers “can’t see” (or don’t care about) things like conics or the sextactic locus. That’s because they want to act on the space by the full group of local diffeomorphisms (an infinite-dimensional group).

I For enumerative geometry, one acts by the 8-dimensional group PGL(3). The problems get increasingly more intricate as one acquires more and more orbits. Eventually there are infinitely many orbits.

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 How our projects differ

I The differential geometers “can’t see” (or don’t care about) things like conics or the sextactic locus. That’s because they want to act on the space by the full group of local diffeomorphisms (an infinite-dimensional group).

I For enumerative geometry, one acts by the 8-dimensional group PGL(3). The problems get increasingly more intricate as one acquires more and more orbits. Eventually there are infinitely many orbits.

I In the differential-geometric setting, it’s not obvious whether the orbit space is finite (at each fixed level) or if there are moduli. In fact there are moduli, but when and where do they enter?

Maunderings in enumerative geometry I It’s a piecewise linear or skeletonized version of algebraic geometry.

I Named in honor of Imre Simon (1943–2009)

http://www.ime.usp.br/

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 Tropical geometry

I A new subject tropical geometry has emerged out of discrete math, optimization, and computer science.

Maunderings in enumerative geometry I Named in honor of Imre Simon (1943–2009)

http://www.ime.usp.br/

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 Tropical geometry

I A new subject tropical geometry has emerged out of discrete math, optimization, and computer science.

I It’s a piecewise linear or skeletonized version of algebraic geometry.

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 Tropical geometry

I A new subject tropical geometry has emerged out of discrete math, optimization, and computer science.

I It’s a piecewise linear or skeletonized version of algebraic geometry.

I Named in honor of Imre Simon (1943–2009)

http://www.ime.usp.br/

Maunderings in enumerative geometry I Each is made up of line segments and rays (edges).

I Each slope is rational.

I Where edges meet there is a balancing condition.

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 Tropical plane curves

I Here are a tropical line and a tropical conic in the plane.

Mikhalkin, Tropical geometry & its applications

Maunderings in enumerative geometry I Each slope is rational.

I Where edges meet there is a balancing condition.

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 Tropical plane curves

I Here are a tropical line and a tropical conic in the plane.

Mikhalkin, Tropical geometry & its applications

I Each is made up of line segments and rays (edges).

Maunderings in enumerative geometry I Where edges meet there is a balancing condition.

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 Tropical plane curves

I Here are a tropical line and a tropical conic in the plane.

Mikhalkin, Tropical geometry & its applications

I Each is made up of line segments and rays (edges).

I Each slope is rational.

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 Tropical plane curves

I Here are a tropical line and a tropical conic in the plane.

Mikhalkin, Tropical geometry & its applications

I Each is made up of line segments and rays (edges).

I Each slope is rational.

I Where edges meet there is a balancing condition.

Maunderings in enumerative geometry I This number is called the degree of the tropical plane curve.

I Here is a tropical cubic curve.

Sottile, Tropical interpolation

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 Tropical plane curves

I The rays point northeast, west, and south, and there are the same number in each direction. (Convention may be rotated by 180◦.)

Maunderings in enumerative geometry I Here is a tropical cubic curve.

Sottile, Tropical interpolation

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 Tropical plane curves

I The rays point northeast, west, and south, and there are the same number in each direction. (Convention may be rotated by 180◦.)

I This number is called the degree of the tropical plane curve.

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 Tropical plane curves

I The rays point northeast, west, and south, and there are the same number in each direction. (Convention may be rotated by 180◦.)

I This number is called the degree of the tropical plane curve.

I Here is a tropical cubic curve.

Sottile, Tropical interpolation

Maunderings in enumerative geometry I Tropicalize it by replacing all multiplications by ⊗ and additions by ⊕, where

I ⊗ means + I ⊕ means “take the minimum.”

I trop(p) = min{a + 2x, b + x + y, c + 2y, d + y, e, f + x}

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 Tropicalizing

I Start with a polynomial, say p(x, y) = ax2 + bxy + cy 2 + dy + e + fx.

Maunderings in enumerative geometry I trop(p) = min{a + 2x, b + x + y, c + 2y, d + y, e, f + x}

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 Tropicalizing

I Start with a polynomial, say p(x, y) = ax2 + bxy + cy 2 + dy + e + fx. I Tropicalize it by replacing all multiplications by ⊗ and additions by ⊕, where

I ⊗ means + I ⊕ means “take the minimum.”

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 Tropicalizing

I Start with a polynomial, say p(x, y) = ax2 + bxy + cy 2 + dy + e + fx. I Tropicalize it by replacing all multiplications by ⊗ and additions by ⊕, where

I ⊗ means + I ⊕ means “take the minimum.”

I trop(p) = min{a + 2x, b + x + y, c + 2y, d + y, e, f + x}

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 Tropicalizing

I Assuming 2b < a + c, 2d < e + c, 2f < a + e, here’s the graph. (Figure from Maclagan & Sturmfels)

I It’s linear on large regions, away from the locus where two (or more) of the six functions tie for achieving the minimum. This locus is the tropical curve defined by trop(p).

Maclagan & Sturmfels

Maunderings in enumerative geometry I This is true, if you can properly interpret the notion: for example, as the degree of the normal bundle.

I B´ezout’sTheorem: Curves of degrees d and e meet in de points. 2 I In particular a curve of degree d has self-intersection d .

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 Classical facts about plane curves

(d−1)(d−2) I The genus of a curve of degree d is 2 .

Maunderings in enumerative geometry I This is true, if you can properly interpret the notion: for example, as the degree of the normal bundle.

2 I In particular a curve of degree d has self-intersection d .

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 Classical facts about plane curves

(d−1)(d−2) I The genus of a curve of degree d is 2 .

I B´ezout’sTheorem: Curves of degrees d and e meet in de points.

Maunderings in enumerative geometry I This is true, if you can properly interpret the notion: for example, as the degree of the normal bundle.

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 Classical facts about plane curves

(d−1)(d−2) I The genus of a curve of degree d is 2 .

I B´ezout’sTheorem: Curves of degrees d and e meet in de points. 2 I In particular a curve of degree d has self-intersection d .

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 Classical facts about plane curves

(d−1)(d−2) I The genus of a curve of degree d is 2 .

I B´ezout’sTheorem: Curves of degrees d and e meet in de points. 2 I In particular a curve of degree d has self-intersection d .

I This is true, if you can properly interpret the notion: for example, as the degree of the normal bundle.

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 Tropical versions

I Tropical B´ezout

Cover of Mathematics Magazine, June 2009

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 Tropical versions

I Tropical genus formula: by graph theory or Euler (d−1)(d−2) characteristic, the number of bounded polygons is 2 .

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 Tropical versions

I Tropical self-intersection: in this example, there are supposed to be 16 self-intersection points. Where are they?

Maunderings in enumerative geometry I The moral: tropical geometry may make your life easier!

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 Tropical versions

I Idea of the proof: jiggle the picture

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 Tropical versions

I Idea of the proof: jiggle the picture

I The moral: tropical geometry may make your life easier!

Maunderings in enumerative geometry