Topics in Algebraic Geometry

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Topics in Algebraic Geometry
By Abraham Taicher Thomas Murphy With Amanda
Knecht (grad) Brendan Hassett (prof)
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Definition Complex Plane Curve

• A Complex Plane Curve is the set of points in
  C2 where a non-constant polynomial vanishes.

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Definition Singularity

• A Singularity is a point p(x0,y0) on a complex
  plane curve Z(f) with all of the following
  satisfied
• f(p)0
• (p)0
• (p)0

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Example singularity

• Fxy-x6-y6
• y-6x5
• x-6y5
• p(0,0) is therefore singular because all
  three conditions are satisfied.
• This singularity is called an ordinary double
  point or node.

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Example Another Singularity

• Fx2yxy2-x4-y4
• y22xy-4x3
• x22xy-4y3
• Again p(0,0) is the only point that satisfies
  all three conditions.
• This singularity is called an ordinary triple
  point.

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Definition Multiplicity

• The multiplicity at the origin µ(f) is the
  order of the lowest degree term of f.
• Examples µ(xy-x6-y6)2
• µ(x2yxy2-x4-y4)3
• Notice that µ(f)lt2 ? f is nonsingular at the
  origin.

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Blowing Up Singularities

• Blowing up is used to parameterize singular
  curves by nonsingular ones.
• A singularity is resolved when it is blown up
  enough times to give a nonsingular curve with
  normal crossings.

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Resolving Singularities Blowing Up

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Example Blowup Final graph
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Differential Forms

• Differential forms dx?dy follow two rules
• dx?dx0
• d(uy)ydu?udy
• We can trace the progression of differential
  forms through the process of blowing up.

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Example Differential Forms

• Fx2-y3
• Blowup 1
• Substitute xuy
• Blowup 2
• Substitute yau
• Blowup 3
• Substitute abu

• dx?dy
• gt (yduudy)?dy
• ydu?dy
• gt audu?(aduuda)
• u2adu?da
• gt u2budu?(bduudb)
• u4bdu ? db

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Properties Log Canonical Threshold

• To calculate the log canonical threshold a, we
  look at the following relation
• Ci is the multiplicity of the exceptional
  divisor after the ith blowup.
• Ki is the multiplicity of the exceptional
  divisor of the ith blowup of the differential
  forms.

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The Problem

• What is the Log Canonical Threshold for
• fxp-yq ?
• Methods
• Combinatorics involving continued fractions.
• The geometry of blowups and the adjunction
  formula.

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ITS OVER!
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