Post on 01-Jun-2020
CMA seminarUniversity of Oslo
11.March 2010
An easy introduction toAlgebraic Geometry
andRational Cuspidal Plane Curves
Torgunn Karoline MoeCMA/MATH
University of Oslo
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
The most interesting objects in the world
• How many and what kind of cusps can a rationalcuspidal plane curve have?
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
This happens today
• Basic algebraic geometry
• Plane algebraic curves
• Singularity theory
• Rational cuspidal plane curves
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Algebraic geometry in a nutshell
• The study of geometric objects using algebraicmethods.
• Can find new and surprising properties of both theobjects and the methods.
• Main tool: commutative algebra.• Rings• Ideals
• Main objects: varieties.• Curves• Surfaces
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
The worlds we work within
• Algebraically closed fields – C.
• Affine spaces of dimension n – Cn,(x1, . . . , xn).
• Projective spaces of dimension n – PnC,
(x0 : . . . : xn).
• Pn can be constructed using Cn+1, identifying pointsin the affine space lying on the same line through theorigin.
Pn ∼= (Cn+1 r {(0, . . . , 0)})/ ∼,
(a0 : . . . : an)∼(λa0 : . . . : λan), ∀ λ ∈ C∗.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
The objects we work with
• An algebraic set in Cn is the zero set V of a finiteset of polynomials in the ring C[x1, . . . , xn].
• An algebraic set in Pn is the zero set V of a finite setof homogeneous polynomials in C[x0, . . . , xn].
• In our worlds open sets are complements of algebraicsets.
• An affine variety is a closed subset of Cn which cannot be decomposed into smaller, closed subsets.
• A projective variety is an irreducible closed subset ofPn.
• An algebraic set in a space of dimension n defined bya single irreducible (homogeneous) polynomial is ahypersurface – a variety of dimension n − 1.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Let’s get it all down to earth
• The projective plane P2 has coordinates (x : y : z).
• One irreducible homogeneous polynomial F (x , y , z)defines V(F ) – a curve in P2.
• The degree of the curve is the degree of thepolynomial.
• Letting z = 1, the polynomial f (x , y) = F (x , y , 1)will define a curve V(f (x , y)) in a space isomorphicto C2.
• Letting y = 1, we get the curve V(f (x , z)) inanother affine plane.
• Letting x = 1, we get V(f (y , z)).
• These three affine curves constitute the projectivecurve V(F ).
• Technically, we have covered P2 by three open affinesets isomorphic to C2,
P2 r V(z), P2 r V(y), P2 r V(x).
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
A typical conic I - V(x2 + y2 − z2)
z = 1 y = 1 x = 1
• Remember that this is just the real picture - thecomplex world hides its secrets.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
A typical conic II - V(x2 + y2+(z − 1)2)
• All conic curves in P2 (circles, ellipses, hyperbolasand parabolas) are equivalent when we work over C.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
The nodal cubic - V(zx2 − zy2 − x3)
• This curve has one obviously interesting point in(0 : 0 : 1).
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
The cuspidal cubic - V(zy2 − x3)
• This curve also has an interesting point in (0 : 0 : 1),but it is different from the point in the previousexample.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Let’s compare
The nodal cubic The cuspidal cubic
• How and why are these interesting and different -even over C?
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Some more theory I
• A point a = (a0 : a1 : a2) on a curve V(F ) is calledsingular if it is in the zero set of all the partialderivatives of F ,
V(Fx(a),Fy (a),Fz(a)).
• A curve can only have a finite number of singularpoints.
• The other points on the curve are called smooth.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Some more theory II
• Every smooth point a on a curve has a uniqe tangentline given by
V(Fx(a)x + Fy (a)y + Fz(a)z).
• Every singular point p has one or more tangentline(s).
• For p = (0 : 0 : 1) singular,
F (x , y , 1) = fm(x , y) + fm+1(x , y) + . . . + fd(x , y).
• The tangent line(s) of C at p is given by the zeroset(s) of each reduced linear factor of fm(x , y).
• A tangent line is special because it touches the curveat the given point a bit more than other lines.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
More about the interesting points
• Unbelieveably many different kinds of singularities.
• Can classify singularities using invariants:• Branches – counting the number of times the curve
passes through the point.• A singularity with more than one branch is called a
multiple point.• A singularity with only one branch is called a cusp.
• Multiplicity – the amount of intersection between ageneral line and the curve at the point.
• Is equal to the m in fm(x , y) for p = (0 : 0 : 1).
• Tangent intersection – the intersection multiplicityof the tangent line and the curve at the point.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Multiplicity 2
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Detonating the algebraic bomb
• Can investigate the inside of a singularity by blowingit up.
• Replace the singularitiy with a projective line.
• In an affine neighbourhood of the singularity, look atall the lines through the point.
• Lift each line to a height corresponding to the slopeof the line.
• Observe that the curve is practically unchangedoutside the singularity.
• Close the curve and get a new curve.
• Look at the point(s) of the new curve correspondingto the blown up singularity.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Blowing up the cuspidal cubic
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
• Yes, the blown up space is strange and funny. And it doesn’t reallylook like that. But don’t worry.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Useful properties of a cusp
• When a cusp is blown up, we have only one pointcorresponding to the singularitiy.
• This point might still be singular.
• Then we blow up again.
• Let mi denote the multiplicity of the remainingsingularity after i blowing-ups.
• For a cusp we define the multiplicity sequence m• m = (m,m1, . . . ,ms).• Have m ≥ m1 ≥ . . . ≥ ms .• There are more restrictions here.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Let’s go back to start
• How many and what kind of cusps can a rationalcuspidal curve have?
• A curve is called cuspidal if all its singular points arecusps.
• A curve of degree d is rational ⇐⇒(d − 1)(d − 2)
2=
∑singular points
(∑
i
mi (mi − 1)
2).
• A rational curve can be given by a parametrization.
• By the formula, the cuspidal cubic is the onlyrational cuspidal curve of degree 3.
• A rational cuspidal plane curve of degree d must alsosatisfy
• Bezout: mp + mq ≤ d .• Matsuoka–Sakai: d < 3 ·m,
where m is the highest multiplicity of the cusps.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Rational cuspidal curves of degree 4
(2), (2), (2) (22), (2) (23)
(3)
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Rational cuspidal curves of degree 5
# Cusps Curve Cuspidal configuration # Curves
1C1 (4) 3 – ABCC2 (26) 1
2C3 (3, 2), (22) 2 – ABC4 (3), (23) 1C5 (24), (22) 1
3C6 (3), (22), (2) 1C7 (22), (22), (22) 1
4 C8 (23), (2), (2), (2) 1
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Conjecture [Piontkowski (2007)]
• There is only one rational cuspidal plane curve withmore than three cusps – the curve of degree 5 withcuspidal configuration [(23), (2), (2), (2)].
• The only tricuspidal curves are• [Fenske, Flenner & Zaidenberg (1996-1999)]
Series d mp mq mr For dI d (d − 2) (2a) (2d−2−a) d ≥ 4II 2a + 3 (d − 3, 2a) (3a) (2) d ≥ 5III 3a + 4 (d − 4, 3a) (4a, 22) (2) d ≥ 7
• The curve of degree 5 with cuspidal configuration[(22), (22), (22)].
Result [Tono (2005)]
• A rational cuspidal curve has ≤ 8 cusps.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
A world of opportunities for me
• Use Cremona transformations to give newrestrictions.
• Construct cuspidal curves by projecting a rationalsmooth curve in Pn.
• There is a connection between the number of cuspsand the centre of projection that is used.
• This is linked to the tangents of the smooth curve inPn.
• Try to interpret the problem in other worlds; i.e.toric geometry or tropical geometry.
Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Useful literature
T. FenskeRational cuspidal plane curves of type (d , d − 4) withχ(ΘV 〈D〉) ≤ 0.
H. Flenner, M. ZaidenbergOn a class of rational cuspidal plane curves.
H. Flenner, M. ZaidenbergRational cuspidal plane curves of type (d , d − 3).
R. HartshorneAlgebraic Geometry.
M. Namba.Geometry of projective algebraic curves.
J. Piontkowski.On the Number of the Cusps of Rational CuspidalPlane Curves.
I hope there’s more cake!