enumerative geometry, intersection theory and …for any given enumerative question three problems...

Enumerative geometry, Intersection Theory andModuli Spaces

Enrico Arbarello

Pisa, May 28, 2008

Euclid (325-265 a.C.)

”Euclid-Bramante”

Postulate: There is one linethrough two distinct points.

Proposition: Two distinct linesmeet in at most one point.

Euclid (325-265 a.C.) ”Euclid-Bramante”

Postulate: There is one linethrough two distinct points.

Postulate: There is oneline through two distinct points.

Apollonius from Perga(262-190 a.C.)

”Conics”

387 Propositions on conics:

”Conics”

Proposition: Through fivegeneric points passes exactlyone conic.

Proposition: Two genericconics meet in at most fourpoints.

Proposition: Through fivegeneric points passes exactlyone conic.

Proposition: Two genericconics meet in at most fourpoints.

Proposition: There are, at most, 4 lines that are tangent to twogiven conics.

Proposition: there are, at most, 8 circles that are tangent to threegiven circles.

I.Newton(1643-1727)

G. Leibniz(1646-1716)

I.Newton(1643-1727)

G. Leibniz(1646-1716)

M.Chasles (1793-1880)

There are (at most)3264 = 51× 24 conics that aretangent to 5 given conics.The difficulty comes fromdouble lines. These solutionsmust be discarded.

H. Schubert (1848-1911)

There are (at most)666.841.080 =23 × 3× 5× 653× 853 spacequadrics which are tanget to 9given quadrics.

M.Chasles (1793-1880)

There are (at most)3264 = 51× 24 conics that aretangent to 5 given conics.

The difficulty comes fromdouble lines. These solutionsmust be discarded.

H. Schubert (1848-1911)

M.Chasles (1793-1880)

H. Schubert (1848-1911)

M.Chasles (1793-1880)

H. Schubert (1848-1911)

M.Chasles (1793-1880)

H. Schubert (1848-1911)

R. Descartes(1596-1650)

line: ax + by + c = 0

circle: x2 + y2 = 1 cubic: y2 = x3 − 1

circle: x2 + y2 = 1

cubic: y2 = x3 − 1

circle: x2 + y2 = 1

cubic: y2 = x3 − 1

quartic: x4 + y4 = 1

nodal cubic: y2 = x3 + x2

quartic: x4 + y4 = 1

nodal cubic: y2 = x3 + x2

Proof of the fact that there are at most 4 lines that aretangent to two given conics:

The analytic represention of a conic shows that the tangent linesto a conic form a conic in the dual plane.

Thus the assertion about tangent lines reduces to the one aboutthe inersection of two conics.

Proof of Apollonius’ 8 Circles Theorem:

The equation of a circle:

x2 + y2 + ax + by + c = 0 , (con (a/2)2 + (b/2)2 − c > 0 ) .

depends on 3 real parameters: a, b e c. These are called themoduli of a circle

The tangency condition is quadratic in the variables a, b e c .

Thus, to find the circles that are tangent to 3 given circles onemust find the common solutions to 3 quadratic equations in thevariables a, b and c .

In general one obtains at most 8 solutions.Q.E.D.

x2 + y2 + ax + by + c = 0 ,

(con (a/2)2 + (b/2)2 − c > 0 ) .

x2 + y2 + ax + by + c = 0 , (con (a/2)2 + (b/2)2 − c > 0 ) .

In general one obtains at most 8 solutions.

Q.E.D.9

x2 + y2 + ax + by + c = 0 , (con (a/2)2 + (b/2)2 − c > 0 ) .

For any given enumerative question three problems naturally arise.

1. Find the natural moduli space M.

2. Study the intersection theory of M.

3. Translate the enumerative problem into an intersectionproblem.

Only the last problem is usually easy to solve. The difficulties lie insolving the first two problems. As far as the second problem isconcerned, the fact that M is a moduli space is, in a sense,irrelevant. One would like to know the intersection theory of everyvariety.

Only the last problem is usually easy to solve. The difficulties lie insolving the first two problems.

As far as the second problem isconcerned, the fact that M is a moduli space is, in a sense,irrelevant. One would like to know the intersection theory of everyvariety.

Looking for missing intersections (I):

Leon Battista Alberti(1404-1472)”De Pictura”

Piero della Francesca(1412- 1492)”De prospectiva pingendi”

Abrecht Durer(1471-1528)”Unterweisung der Messungmit dem Zirkel undRichtscheit”

With projective geometry a perfect duality is established betweenpoints and lines in the plane.

The sentence: there is onlyone line through two distinctpoints, is perfectly dual to: two

distinct lines intersect inexactly one point.

The sentence: there is onlyone line through two distinctpoints,

is perfectly dual to: twodistinct lines intersect inexactly one point.

Looking for missing intersections (II):

( 3, 0 )

(0, 2) Circle: x2 + y2 = 3Line: y = 2Intersection points:x2 + 22 = 3, that isx2 + 4 = 3, that is x2 = −1.

√−1 = i

Intersection points: (i , 2) e (−i , 2).

( 3, 0 )

(0, 2)

Circle: x2 + y2 = 3Line: y = 2Intersection points:x2 + 22 = 3, that isx2 + 4 = 3, that is x2 = −1.

√−1 = i

( 3, 0 )

√−1 = i

( 3, 0 )

√−1 = i

( 3, 0 )

√−1 = i

Numbers:1, 2, 3, . . .

0, 1, 2, 3, . . .

· · · − 3,−2,−1, 0, 1, 2, 3, . . .

x2 = 2

The real line R:

Numbers:1, 2, 3, . . .

0, 1, 2, 3, . . .

· · · − 3,−2,−1, 0, 1, 2, 3, . . .

x2 = 2

The real line R:

Numbers:1, 2, 3, . . .

0, 1, 2, 3, . . .

· · · − 3,−2,−1, 0, 1, 2, 3, . . .

x2 = 2

The real line R:

Numbers:1, 2, 3, . . .

0, 1, 2, 3, . . .

· · · − 3,−2,−1, 0, 1, 2, 3, . . .

x2 = 2

The real line R:

Numbers:1, 2, 3, . . .

0, 1, 2, 3, . . .

· · · − 3,−2,−1, 0, 1, 2, 3, . . .

x2 = 2

The real line R:

Numbers:1, 2, 3, . . .

0, 1, 2, 3, . . .

· · · − 3,−2,−1, 0, 1, 2, 3, . . .

x2 = 2

The real line R:

Numbers:1, 2, 3, . . .

0, 1, 2, 3, . . .

· · · − 3,−2,−1, 0, 1, 2, 3, . . .

x2 = 2

The real line R:

.. . . ....π21 30 2-1

.5 2-8 3

Nicolo Tartaglia(Brescia 1500 - Venezia 1557)

Gerolamo Cardano(1515 Pavia - Roma 1576)

”Ars Magna”

Raffaele Bombelli(Bologna 1526 - Roma 1572)”Algebra”

Omar KhayyamGhiyath al-Din Abu’l-FathUmar ibn Ibrahim Al-Nisaburial-Khayyami(Nishapur 1048 - 1131)

”Ars Magna”

Caspar Wessel(1745 - 1818)Geometrical interpretation ofcomplex numbers.

The complex plane C:Instead of a real variable x , one considers a complex variablez = u + iv and instead of the real variable y , one considers acomplex variable w = ξ + iη. What happens to plane curves?

The complex plane C:

Instead of a real variable x , one considers a complex variablez = u + iv and instead of the real variable y , one considers acomplex variable w = ξ + iη. What happens to plane curves?

The complex plane C:

Instead of a real variable x , one considers a complex variablez = u + iv and instead of the real variable y , one considers acomplex variable w = ξ + iη. What happens to plane curves?

Descartes:

A plane algebraic curve isgiven by a polynomial equationP(x , y) = 0 , in the real planewith coordinates x e y .

For example:

line: P(x , y) = x − y + 1circle: P(x , y) = x2 + y2 − 1cubic : P(x , y) = x3 − x2 − y2

What happens in the complexplane with complexcoordinates z e w ?

Descartes:

For example:

Descartes:

For example:

P(x,y)=0

Descartes:

For example:

P(x,y)=0

Descartes:

For example:

P(x,y)=0

P(z, w)=0 ?

z=u+iv

w=s+it

Bernhard Riemann(Breselenz [Hanover] 1826- Selasca [Como] 1866)

x − 23y − 1 = 0

z − 23w − 1 = 0

x2 + y2 = 1 z2 + w2 = 1

x − 23y − 1 = 0

z − 23w − 1 = 0

x2 + y2 = 1 z2 + w2 = 1

x − 23y − 1 = 0

z − 23w − 1 = 0

x2 + y2 = 1

z2 + w2 = 1

x − 23y − 1 = 0

z − 23w − 1 = 0

x2 + y2 = 1 z

z2 + w2 = 1

y2 = x3 − x

w2 = z3 − z

x4 + y4 = 1 z4 + w4 = 1

y2 = x3 − x

w2 = z3 − z

x4 + y4 = 1 z4 + w4 = 1

y2 = x3 − x

w2 = z3 − z

x4 + y4 = 1

z4 + w4 = 1

y2 = x3 − x

w2 = z3 − z

x4 + y4 = 1 z

z4 + w4 = 1

In conclusion, in the complex projective plane P2C, with complexcoordinates z and w , the solutions to the equation P(z ,w) = 0form a surface which is called a Rieman surface. The number of”holes” of this surface is called the genus of the surface and it isdenoted with the letter g.

w........... P(z, w)=0

Riemann pulled algebraic curves out of their ambient space andmade them live in thin air.

So, when considering the cubic curve

Γ : y2 = x3 − 1 ,

one is really looking at an abstract, genus 1, compact Riemannsurface C and at pair of meromorphic functions x , y ∈M(C )realizing the embedding of C in PC2 (the poles of x and y go theline at infinity). The Riemann surface C is just a compact analyticmanifold of dimension 1.

(x,y)C Γ

Riemann pulled algebraic curves out of their ambient space andmade them live in thin air. So, when considering the cubic curve

Γ : y2 = x3 − 1 ,

(x,y)C Γ

Γ : y2 = x3 − 1 ,

one is really looking at an abstract, genus 1, compact Riemannsurface C and at pair of meromorphic functions x , y ∈M(C )realizing the embedding of C in PC2 (the poles of x and y go theline at infinity).

The Riemann surface C is just a compact analyticmanifold of dimension 1.

(x,y)C Γ

Γ : y2 = x3 − 1 ,

(x,y)C Γ

The Riemann surface C is the correct ambient space to dofunction theory in one variable, i.e. the theory of holomorphic andmeromorphic functions in one variable.

Choosing r meromorphicfunctions on C yields an analytic map from C to PCr andtherefore a projective realization of C . For instance, in the exampleabove, we could have looked only at the function x , then we wouldbe realizing C as a 2-sheeted cover of PC1. Looking at y , on theother hand, is like presenting C as a 3-sheeted cover of PC1.Bringing into the picture a third function z would map C to PC4

realizing it, perhaps, as a smooth quartic curve.From this non-cartesian point of view, algebraic equations are onlylinked to the accidental projective manifestations of a Riemannsurface and they position themselves in the background.

The Riemann surface C is the correct ambient space to dofunction theory in one variable, i.e. the theory of holomorphic andmeromorphic functions in one variable. Choosing r meromorphicfunctions on C yields an analytic map from C to PCr andtherefore a projective realization of C .

For instance, in the exampleabove, we could have looked only at the function x , then we wouldbe realizing C as a 2-sheeted cover of PC1. Looking at y , on theother hand, is like presenting C as a 3-sheeted cover of PC1.Bringing into the picture a third function z would map C to PC4

The Riemann surface C is the correct ambient space to dofunction theory in one variable, i.e. the theory of holomorphic andmeromorphic functions in one variable. Choosing r meromorphicfunctions on C yields an analytic map from C to PCr andtherefore a projective realization of C . For instance, in the exampleabove, we could have looked only at the function x , then we wouldbe realizing C as a 2-sheeted cover of PC1.

Looking at y , on theother hand, is like presenting C as a 3-sheeted cover of PC1.Bringing into the picture a third function z would map C to PC4

The Riemann surface C is the correct ambient space to dofunction theory in one variable, i.e. the theory of holomorphic andmeromorphic functions in one variable. Choosing r meromorphicfunctions on C yields an analytic map from C to PCr andtherefore a projective realization of C . For instance, in the exampleabove, we could have looked only at the function x , then we wouldbe realizing C as a 2-sheeted cover of PC1. Looking at y , on theother hand, is like presenting C as a 3-sheeted cover of PC1.

Bringing into the picture a third function z would map C to PC4

The Riemann surface C is the correct ambient space to dofunction theory in one variable, i.e. the theory of holomorphic andmeromorphic functions in one variable. Choosing r meromorphicfunctions on C yields an analytic map from C to PCr andtherefore a projective realization of C . For instance, in the exampleabove, we could have looked only at the function x , then we wouldbe realizing C as a 2-sheeted cover of PC1. Looking at y , on theother hand, is like presenting C as a 3-sheeted cover of PC1.Bringing into the picture a third function z would map C to PC4

realizing it, perhaps, as a smooth quartic curve.

From this non-cartesian point of view, algebraic equations are onlylinked to the accidental projective manifestations of a Riemannsurface and they position themselves in the background.

The Riemann surface C is the correct ambient space to dofunction theory in one variable, i.e. the theory of holomorphic andmeromorphic functions in one variable. Choosing r meromorphicfunctions on C yields an analytic map from C to PCr andtherefore a projective realization of C . For instance, in the exampleabove, we could have looked only at the function x , then we wouldbe realizing C as a 2-sheeted cover of PC1. Looking at y , on theother hand, is like presenting C as a 3-sheeted cover of PC1.Bringing into the picture a third function z would map C to PC4

Now that algebraic curves are pulled out from an ambient space, itis their intrinsic geometry that comes in the forefront.

From manypoints of view, two Riemann surfaces that are bianalyticallyequivalent (i.e. isomorphic) are indistinguishable. For instance theyhave exactly the same projective realizations.

Mg = {Riemann surfaces of genus g}/isomorphisms

This moduli space exists and it is an essentially smooth, complexvariety of dimension 3g − 3. (There is an exception when thegenus is 1. In fact, in order to prevent the occurrence of infinitelymany automorphisms one should always choose a point on a genus1 Riemann surface and insist that any automorphism should keepthat marked point fixed.)

Now that algebraic curves are pulled out from an ambient space, itis their intrinsic geometry that comes in the forefront. From manypoints of view, two Riemann surfaces that are bianalyticallyequivalent (i.e. isomorphic) are indistinguishable. For instance theyhave exactly the same projective realizations.

This moduli space exists and it is an essentially smooth, complexvariety of dimension 3g − 3.

(There is an exception when thegenus is 1. In fact, in order to prevent the occurrence of infinitelymany automorphisms one should always choose a point on a genus1 Riemann surface and insist that any automorphism should keepthat marked point fixed.)

In some sense, the moduli space Mg plays the same role as themoduli space of circles that we saw before.

To solve anenumerative problem concerning circles, we translated this problemin an intersection problem in the moduli space. Thus one canimagine that to solve enumerative problems concerning algebraiccurves one should have a good understanding of the intersectiontheory of Mg .

In some sense, the moduli space Mg plays the same role as themoduli space of circles that we saw before. To solve anenumerative problem concerning circles, we translated this problemin an intersection problem in the moduli space.

Thus one canimagine that to solve enumerative problems concerning algebraiccurves one should have a good understanding of the intersectiontheory of Mg .

In some sense, the moduli space Mg plays the same role as themoduli space of circles that we saw before. To solve anenumerative problem concerning circles, we translated this problemin an intersection problem in the moduli space. Thus one canimagine that to solve enumerative problems concerning algebraiccurves one should have a good understanding of the intersectiontheory of Mg .

A quick way to see why the isomorphism class of a compactRiemann surface of genus g > 1 depends on 3g − 3 complexparameters, or 6g − 6 real ones, is to decompose the surface in2g − 2 pants by suitably cutting it along 3g − 3 curves:

We can endow, in a unique way, the Riemann surface C with acomplete hyperbolic metric with constant curvature equal to -1,and we can assume that the 3g − 3 dissecting curves are geodesics.Their lengths provide 3g − 3 real moduli. Once these are fixed, thevarious pants are analytically rigid and the only free parametersinvolved in reconstructing C from them are the 3g − 3 twists thatwe can perform when identifying their boundary curves. These arethe additional 3g − 3 real moduli.

We can endow, in a unique way, the Riemann surface C with acomplete hyperbolic metric with constant curvature equal to -1,and we can assume that the 3g − 3 dissecting curves are geodesics.

Their lengths provide 3g − 3 real moduli. Once these are fixed, thevarious pants are analytically rigid and the only free parametersinvolved in reconstructing C from them are the 3g − 3 twists thatwe can perform when identifying their boundary curves. These arethe additional 3g − 3 real moduli.

We can endow, in a unique way, the Riemann surface C with acomplete hyperbolic metric with constant curvature equal to -1,and we can assume that the 3g − 3 dissecting curves are geodesics.Their lengths provide 3g − 3 real moduli.

Once these are fixed, thevarious pants are analytically rigid and the only free parametersinvolved in reconstructing C from them are the 3g − 3 twists thatwe can perform when identifying their boundary curves. These arethe additional 3g − 3 real moduli.

We can endow, in a unique way, the Riemann surface C with acomplete hyperbolic metric with constant curvature equal to -1,and we can assume that the 3g − 3 dissecting curves are geodesics.Their lengths provide 3g − 3 real moduli. Once these are fixed, thevarious pants are analytically rigid and the only free parametersinvolved in reconstructing C from them are the 3g − 3 twists thatwe can perform when identifying their boundary curves.

These arethe additional 3g − 3 real moduli.

Let us go back to a plane curve given by an equation P(z ,w) = 0.How does one compute the genus of a plane curve?

Examples:

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

curve of degree d : zd + wd = 1 g =(d − 1)(d − 2)

A curve of genus 0 is the natural generalization of a line.At first sight, it would seem that the only genus zero curves arethe lines and the conics.But it is not so. This depends on the fact that a plane curve mayhave ”nodes”. Let us look at a cubic:

Let us go back to a plane curve given by an equation P(z ,w) = 0.How does one compute the genus of a plane curve?Examples:

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

A curve of genus 0 is the natural generalization of a line.

At first sight, it would seem that the only genus zero curves arethe lines and the conics.But it is not so. This depends on the fact that a plane curve mayhave ”nodes”. Let us look at a cubic:

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

A curve of genus 0 is the natural generalization of a line.At first sight, it would seem that the only genus zero curves arethe lines and the conics.

But it is not so. This depends on the fact that a plane curve mayhave ”nodes”. Let us look at a cubic:

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

A curve of genus 0 is the natural generalization of a line.At first sight, it would seem that the only genus zero curves arethe lines and the conics.But it is not so.

This depends on the fact that a plane curve mayhave ”nodes”. Let us look at a cubic:

line : z + w = 1 g = 0 ,

conic : z2 + w2 = 1 g = 0 ,

cubic : z3 + w3 = 1 g = 1 ,

quartic : z4 + w4 = 1 g = 3 ,

y2 = x3 − x

w2 = z3 − z

y2 = x3 − x2 w2 = z3 − z2

y2 = x3 − x

w2 = z3 − z

y2 = x3 − x2 w2 = z3 − z2

y2 = x3 − x

w2 = z3 − z

y2 = x3 − x2

w2 = z3 − z2

y2 = x3 − x

w2 = z3 − z

y2 = x3 − x2 z

w2 = z3 − z2

A cubic with one node should be considered as a surface of genus0 and not of genus 1:

In an analogous way a surface like:

should be considered of genus3 and not of genus 5!

In concusion: a degree d plane curve with δ nodes should beconsidered as a Riemann surface of genus

g =(d − 1)(d − 2)

2− δ .

Thus there are many genus 0 curves ! These are all degree d theplane curves with

δ =(d − 1)(d − 2)

nodes. We now treat genus 0 curves in the way Euclid andApollonius were treating lines and conics. We can immediately seeone difference. Whereas, according to Euclid, there is a uniqueline through 2 distinct points and, according to Apollonius, there isa unique conic through 5 points in general position, things are notso simple for genus 0 curves of degree d , as soon as d > 2. Theonly thing that one can say without difficulty is that there is afinite number of degree d , genus 0, plane curves passing through3d − 1 points in general position. Let Nd be this number. How tocompute Nd?

g =(d − 1)(d − 2)

2− δ .

δ =(d − 1)(d − 2)

g =(d − 1)(d − 2)

2− δ .

Thus there are many genus 0 curves !

These are all degree d theplane curves with

δ =(d − 1)(d − 2)

g =(d − 1)(d − 2)

2− δ .

δ =(d − 1)(d − 2)

nodes.

We now treat genus 0 curves in the way Euclid andApollonius were treating lines and conics. We can immediately seeone difference. Whereas, according to Euclid, there is a uniqueline through 2 distinct points and, according to Apollonius, there isa unique conic through 5 points in general position, things are notso simple for genus 0 curves of degree d , as soon as d > 2. Theonly thing that one can say without difficulty is that there is afinite number of degree d , genus 0, plane curves passing through3d − 1 points in general position. Let Nd be this number. How tocompute Nd?

g =(d − 1)(d − 2)

2− δ .

δ =(d − 1)(d − 2)

nodes. We now treat genus 0 curves in the way Euclid andApollonius were treating lines and conics.

We can immediately seeone difference. Whereas, according to Euclid, there is a uniqueline through 2 distinct points and, according to Apollonius, there isa unique conic through 5 points in general position, things are notso simple for genus 0 curves of degree d , as soon as d > 2. Theonly thing that one can say without difficulty is that there is afinite number of degree d , genus 0, plane curves passing through3d − 1 points in general position. Let Nd be this number. How tocompute Nd?

g =(d − 1)(d − 2)

2− δ .

δ =(d − 1)(d − 2)

nodes. We now treat genus 0 curves in the way Euclid andApollonius were treating lines and conics. We can immediately seeone difference.

Whereas, according to Euclid, there is a uniqueline through 2 distinct points and, according to Apollonius, there isa unique conic through 5 points in general position, things are notso simple for genus 0 curves of degree d , as soon as d > 2. Theonly thing that one can say without difficulty is that there is afinite number of degree d , genus 0, plane curves passing through3d − 1 points in general position. Let Nd be this number. How tocompute Nd?

g =(d − 1)(d − 2)

2− δ .

δ =(d − 1)(d − 2)

nodes. We now treat genus 0 curves in the way Euclid andApollonius were treating lines and conics. We can immediately seeone difference. Whereas, according to Euclid, there is a uniqueline through 2 distinct points and, according to Apollonius, there isa unique conic through 5 points in general position,

things are notso simple for genus 0 curves of degree d , as soon as d > 2. Theonly thing that one can say without difficulty is that there is afinite number of degree d , genus 0, plane curves passing through3d − 1 points in general position. Let Nd be this number. How tocompute Nd?

g =(d − 1)(d − 2)

2− δ .

δ =(d − 1)(d − 2)

nodes. We now treat genus 0 curves in the way Euclid andApollonius were treating lines and conics. We can immediately seeone difference. Whereas, according to Euclid, there is a uniqueline through 2 distinct points and, according to Apollonius, there isa unique conic through 5 points in general position, things are notso simple for genus 0 curves of degree d , as soon as d > 2.

Theonly thing that one can say without difficulty is that there is afinite number of degree d , genus 0, plane curves passing through3d − 1 points in general position. Let Nd be this number. How tocompute Nd?

g =(d − 1)(d − 2)

2− δ .

δ =(d − 1)(d − 2)

nodes. We now treat genus 0 curves in the way Euclid andApollonius were treating lines and conics. We can immediately seeone difference. Whereas, according to Euclid, there is a uniqueline through 2 distinct points and, according to Apollonius, there isa unique conic through 5 points in general position, things are notso simple for genus 0 curves of degree d , as soon as d > 2. Theonly thing that one can say without difficulty is that there is afinite number of degree d , genus 0, plane curves passing through3d − 1 points in general position.

Let Nd be this number. How tocompute Nd?

g =(d − 1)(d − 2)

2− δ .

δ =(d − 1)(d − 2)

nodes. We now treat genus 0 curves in the way Euclid andApollonius were treating lines and conics. We can immediately seeone difference. Whereas, according to Euclid, there is a uniqueline through 2 distinct points and, according to Apollonius, there isa unique conic through 5 points in general position, things are notso simple for genus 0 curves of degree d , as soon as d > 2. Theonly thing that one can say without difficulty is that there is afinite number of degree d , genus 0, plane curves passing through3d − 1 points in general position. Let Nd be this number.

How tocompute Nd?

g =(d − 1)(d − 2)

2− δ .

δ =(d − 1)(d − 2)

Thus the misterious number Nd is number of plane curves ofdegree d and genus 0 passing through 3d − 1 given points ingeneral position.

What to we know about Nd?

d 3d − 1 Nd

Euclid (≈300 a.C.) 1 2 N1 = 1

Apollonius (≈240 a.C.) 2 5 N2 = 1

Chasles (≈1820) 3 8 N3 = 12

Schubert (≈1870) 4 11 N4 = 620

Schubert 5 14 N5 = 87304

Why do we care?

Thus the misterious number Nd is number of plane curves ofdegree d and genus 0 passing through 3d − 1 given points ingeneral position. What to we know about Nd?

d 3d − 1 Nd

Euclid (≈300 a.C.) 1 2 N1 = 1

Apollonius (≈240 a.C.) 2 5 N2 = 1

Chasles (≈1820) 3 8 N3 = 12

Schubert (≈1870) 4 11 N4 = 620

Schubert 5 14 N5 = 87304

Why do we care?

d 3d − 1 Nd

Euclid (≈300 a.C.) 1 2 N1 = 1

Apollonius (≈240 a.C.) 2 5 N2 = 1

Chasles (≈1820) 3 8 N3 = 12

Schubert (≈1870) 4 11 N4 = 620

Schubert 5 14 N5 = 87304

Why do we care?

d 3d − 1 Nd

Euclid (≈300 a.C.) 1 2 N1 = 1

Apollonius (≈240 a.C.) 2 5 N2 = 1

Chasles (≈1820) 3 8 N3 = 12

Schubert (≈1870) 4 11 N4 = 620

Schubert 5 14 N5 = 87304

Why do we care?

d 3d − 1 Nd

Euclid (≈300 a.C.) 1 2 N1 = 1

Apollonius (≈240 a.C.) 2 5 N2 = 1

Chasles (≈1820) 3 8 N3 = 12

Schubert (≈1870) 4 11 N4 = 620

Schubert 5 14 N5 = 87304

Why do we care?

d 3d − 1 Nd

Euclid (≈300 a.C.) 1 2 N1 = 1

Apollonius (≈240 a.C.) 2 5 N2 = 1

Chasles (≈1820) 3 8 N3 = 12

Schubert (≈1870) 4 11 N4 = 620

Schubert 5 14 N5 = 87304

Why do we care?

d 3d − 1 Nd

Euclid (≈300 a.C.) 1 2 N1 = 1

Apollonius (≈240 a.C.) 2 5 N2 = 1

Chasles (≈1820) 3 8 N3 = 12

Schubert (≈1870) 4 11 N4 = 620

Schubert 5 14 N5 = 87304

Why do we care?

d 3d − 1 Nd

Euclid (≈300 a.C.) 1 2 N1 = 1

Apollonius (≈240 a.C.) 2 5 N2 = 1

Chasles (≈1820) 3 8 N3 = 12

Schubert (≈1870) 4 11 N4 = 620

Schubert 5 14 N5 = 87304

Why do we care?

d 3d − 1 Nd

Euclid (≈300 a.C.) 1 2 N1 = 1

Apollonius (≈240 a.C.) 2 5 N2 = 1

Chasles (≈1820) 3 8 N3 = 12

Schubert (≈1870) 4 11 N4 = 620

Schubert 5 14 N5 = 87304

Why do we care?

Soliton waves,

J.Scott-Russell (1808-1882)

The ”Scott Russell” Aqueduct, Union Canal, not far fromHeriot-Watt University (July 12, 1995).

Soliton waves,

.u(x,t)

Soliton waves,

.u(x,t)

“ I believe I shall best introduce this phenomenon by describing thecircumstances of my own first acquaintances with it. I was observing themotion of a boat which was rapidly drawn along a narrow channel by apair of horses, when the boat suddenly stopped — not so the mass ofwater in the channel which it had put in motion; it accumulated roundthe prow of the vessel in a state of violent agitation, then suddenlyleaving it behind, rolled forward with great velocity, assuming the form ofa large solitary elevation, a rounded, smooth and well defined heap ofwater, which continued its course along the channel apparently withoutchange of form or diminution of speed. I followed it on horseback andovertook it still rolling on at a rate of some eight or nine miles an hour,preserving its original figure some thirty feet long and a foot to a foot anda half in height. Its height gradually diminished and after a chase of oneor two miles I lost it in the windings of the channel. Such in the monthof August 1834 was my first chance interview with that singular andbeautiful phenomenon which I have called the Wave of Translation...”

Two soliton waves of differentamplitude and speed,interacting and thencontinuing their course withintact profile and unchangedspeed.

The differential equation governing solitary waves was discoveredby two dutch mathematical-physicists Korteweg and de Vriesaround 1890. This equation is known as KdV equation