an easy introduction to algebraic geometry and rational ...folk.uio.no/torgunnk/cma.pdfintroduction...
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CMA seminarUniversity of Oslo
11.March 2010
An easy introduction toAlgebraic Geometry
andRational Cuspidal Plane Curves
Torgunn Karoline MoeCMA/MATH
University of Oslo
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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?
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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
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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
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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∗.
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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.
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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).
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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.
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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.
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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).
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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.
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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?
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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.
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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.
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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.
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Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Multiplicity 2
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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.
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Torgunn Karoline Moe
Cuspidal curves
Introduction
Algebraicgeometry
Algebraiccurves
Singularitytheory
Cuspidal curves
References
Blowing up the cuspidal cubic
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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.
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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.
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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.
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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)
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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
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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.
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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.
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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.
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I hope there’s more cake!