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Counting lines on surfacesCounting rational curves in the plane

Counting rational curves on the quintic hypersurfaceThe Semple tower (a.k.a. the Monster tower)

Tropical curves

Maunderings in enumerative geometry

Gary Kennedy

Ohio State University

UC Santa Cruz, January 2015

Tropical curves

“Maunderings”

Maundering

I “Rambling talk, drivel” (Oxford English Dictionary)

I “A rambling or pointless discourse” (Wiktionary)

Tropical curves

“Maunderings”

Maundering

I “Rambling talk, drivel” (Oxford English Dictionary)

I “A rambling or pointless discourse” (Wiktionary)

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

www.carus-verlag.com/

Franz Schubert (1797–1828) Hermann Schubert (1848–1911)

Tropical curves

www.carus-verlag.com/

Franz Schubert (1797–1828) Hermann Schubert (1848–1911)

Tropical curves

What’s a surface?

I A surface is the set of points in 3-dimensional space whosecoordinates satisfy a single polynomial equation.

I For each term, its degree is the sum of its exponents. Themaximum value is called the degree of the surface.

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What’s a surface?

I A surface is the set of points in 3-dimensional space whosecoordinates satisfy a single polynomial equation.

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A sample problem

I How many lines lie on the degree-2 surfacex2 + y2 − z2 − 1 = 0?

http://www.math.umn.edu/ rogness/quadrics/hyp1sh.gif

Tropical curves

A sample problem

I Answer: infinitely many!

I Example: Slice the surface using the plane x = 1 (the tangentplane at the point (1, 0, 0)). This gives the curve y2 − z2 = 0,a pair of lines.

I The same thing works at every point: if you slice using thetangent plane, you obtain a conic section. But it can’t be ahyperbola or parabola or ellipse; it must be a degenerate type,which means that the equation of the curve can be factoredinto two linear equations.

I Conclusion: this is a doubly ruled surface.

Tropical curves

A sample problem

Tropical curves

A sample problem

Tropical curves

A sample problem

Tropical curves

A caveat

I I make the same claim for the ellipsoid 4x2 + y2 + 4z2− 4 = 0.

www.math.hmc.edu/ gu

I Huh?

Tropical curves

A caveat

I I make the same claim for the ellipsoid 4x2 + y2 + 4z2− 4 = 0.

www.math.hmc.edu/ gu

I Huh?

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 + y2 = 0.

I 4x2 + y2 = (2x + iy)(2x − iy)

I Thus the surface contains two lines through this point.

I Conventional assumption of algebraic geometry: points areallowed to have complex numbers as coordinates, andequations are allowed to have complex numbers as coefficients.

Tropical curves

A caveat

I 4x2 + y2 = (2x + iy)(2x − iy)

Tropical curves

A caveat

I 4x2 + y2 = (2x + iy)(2x − iy)

Tropical curves

A caveat

I 4x2 + y2 = (2x + iy)(2x − iy)

Tropical curves

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.

Tropical curves

The 27 lines on a cubic

www.cubics.algebraicsurface.net

I Example: On the Fermat cubic x3 + y3 + z3 − 1 = 0, here are9 of the 27 lines:

x = α, z = −βy

where α and β are cube roots of 1.

Tropical curves

What’s a rational curve?

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

x =p(t)

r(t), y =

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

Tropical curves

What’s a rational curve?

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

x =p(t)

r(t), y =

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

Tropical curves

A rational cubic

I Example of a rational curve of degree 3

x =t2 − 1

1, y =

t3 − t

I Intrinsic equation: y2 = x3 + x2

Tropical curves

A rational cubic

I Example of a rational curve of degree 3

x =t2 − 1

1, y =

t3 − t

I Intrinsic equation: y2 = x3 + x2

Tropical curves

Rational = genus 0

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

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

I Rational curves are those of genus 0.

Tropical curves

Rational = genus 0

Tropical curves

Rational = genus 0

Tropical curves

Counting rational plane curves

I When specifying a rational curve, you have a lot of freedom inspecifying the rational functions. One can show there are3d − 1 degrees of freedom, where d is the degree.

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

Tropical curves

I When specifying a rational curve, you have a lot of freedom inspecifying the rational functions. One can show there are3d − 1 degrees of freedom, where d is the degree.

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

Tropical curves

d 3d − 1 N(d)

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.

Tropical curves

d 3d − 1 N(d)

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.

Tropical curves

My work with Ernstrom and Colley

I Ernstrom and I asked: what if some of the 3d − 1 “pointconditions” are replaced by “line conditions” (meaning thatthe curve must be tangent to specified lines)?

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

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

Tropical curves

One motivation

I Why study such questions?

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

I The simplest example is the hypersurface of degree 5 in4-dimensional space (called the quintic 3-fold).

Tropical curves

One motivation

Tropical curves

One motivation

Tropical curves

What’s a hypersurface?

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

I Example of a quintic 3-fold (hypersurface of degree 5 in4-dimensional space): w5 + x5 + y5 + z5 − wxyz = 0.

Tropical curves

What’s so special about a quintic 3-fold?

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

I Clemens’s conjecture: This expectation is fulfilled.

Tropical curves

What’s so special about a quintic 3-fold?

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

I Clemens’s conjecture: This expectation is fulfilled.

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 arecounting something slightly different (“instanton numbers”).Their arguments mix rigorous calculations with physicalintuition and sophisticated numerology. Rigorous statementsand proofs have now been given up to degree 10.

Tropical curves

Curvilinear data

I Suppose C is a smooth plane curve passing through theorigin, and that its tangent line there isn’t vertical. Then wecan associate to it a sequence of curvilinear data, the valuesat the origin of

dx,d2y

dx2, . . . ,

I We can also do this for certain singular curves, by takinglimits.

Tropical curves

Curvilinear data

I Suppose C is a smooth plane curve passing through theorigin, and that its tangent line there isn’t vertical. Then wecan associate to it a sequence of curvilinear data, the valuesat the origin of

dx,d2y

dx2, . . . ,

I We can also do this for certain singular curves, by takinglimits.

Tropical curves

Curvilinear data of a singular curve

I Example: (y − x2)2 = x5

I Parametrize using x = t2 and y = t4 + t5.

I Calculate

y ′ =dy

dx= 2t2 +

2t3 → 0 (as t → 0)

y ′′ =dy ′

dx= 2 +

4t → 2

Tropical curves

I Calculate

y ′ =dy

dx= 2t2 +

2t3 → 0 (as t → 0)

y ′′ =dy ′

dx= 2 +

4t → 2

Tropical curves

I Calculate

y ′ =dy

dx= 2t2 +

2t3 → 0 (as t → 0)

y ′′ =dy ′

dx= 2 +

4t → 2

Tropical curves

I This calculation shows that the curve has the same curvilineardata up to second order as the parabola y = x2.

y ′ =dy

dx= 2t2 +

2t3 → 0

y ′′ =dy ′

dx= 2 +

4t → 2

I But at the next step of the calculation,

dy ′′

8t→∞.

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

Tropical curves

y ′ =dy

dx= 2t2 +

2t3 → 0

y ′′ =dy ′

dx= 2 +

4t → 2

dy ′′

8t→∞.

Tropical curves

y ′ =dy

dx= 2t2 +

2t3 → 0

y ′′ =dy ′

dx= 2 +

4t → 2

dy ′′

8t→∞.

Tropical curves

I Here the “obvious fix” is to turn things upside down:

dy ′′=

15→ 0.

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

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

Tropical curves

dy ′′=

15→ 0.

Tropical curves

dy ′′=

15→ 0.

Tropical curves

The Semple tower

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

I It leads to the Semple towerS(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 enumerativegeometry.

Tropical curves

The Semple tower

Tropical curves

The Semple tower

Tropical curves

The Semple tower

Tropical curves

Sextactic points

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

I By counting parameters, we see that the curvilinear data ofpoints 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 algebraicgeometry), 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 are3d(4d − 9) such points.

Tropical curves

Sextactic points

Tropical curves

Sextactic points

Tropical curves

Sextactic points

Tropical curves

Rediscovery

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

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

Tropical curves

Rediscovery

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

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

Tropical curves

How our projects agree

I We do the same constructions.

I We begin to analyze the tower by a coarse classification: toeach point is attached a code word. In the algebro-geometriccode 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 anelementary translation between the codes.

Tropical curves

How our projects agree

I We do the same constructions.

I We begin to analyze the tower by a coarse classification: toeach point is attached a code word. In the algebro-geometriccode 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 anelementary translation between the codes.

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 theywant to act on the space by the full group of localdiffeomorphisms (an infinite-dimensional group).

I For enumerative geometry, one acts by the 8-dimensionalgroup PGL(3). The problems get increasingly more intricateas one acquires more and more orbits. Eventually there areinfinitely many orbits.

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

Tropical curves

Tropical geometry

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

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

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

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

Tropical curves

Tropical geometry

Tropical curves

Tropical geometry

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.

Tropical curves

I The rays point northeast, west, and south, and there are thesame number in each direction. (Convention may be rotatedby 180◦.)

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

I Here is a tropical cubic curve.

Sottile, Tropical interpolation

Tropical curves

Tropicalizing

I Start with a polynomial, sayp(x , y) = ax2 + bxy + cy2 + dy + e + fx .

I Tropicalize it by replacing all multiplications by ⊗ andadditions 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}

Tropical curves

Tropicalizing

Tropical curves

Tropicalizing

Tropical curves

Tropicalizing

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

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

Maclagan & Sturmfels

Tropical curves

Classical facts about plane curves

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

I Bezout’s Theorem: Curves of degrees d and e meet in depoints.

I In particular a curve of degree d has self-intersection d2.

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

Tropical curves

I In particular a curve of degree d has self-intersection d2.I This is true, if you can properly interpret the notion: for

example, as the degree of the normal bundle.

Tropical curves

Tropical versions

I Tropical Bezout

Cover of Mathematics Magazine, June 2009

Tropical curves

Tropical versions

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

Tropical curves

Tropical versions

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

Tropical curves

Tropical versions

I Idea of the proof: jiggle the picture

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

Tropical curves

Tropical versions

I Idea of the proof: jiggle the picture

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

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