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Topics in Algebraic Geometry

Topics in Algebraic Geometry. By: Abraham Taicher Thomas Murphy With: Amanda Knecht (grad) Brendan Hassett (prof). Definition: Complex Plane Curve. A Complex Plane Curve is the set of points in C 2 where a non-constant polynomial vanishes. Definition: Singularity.

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Topics in Algebraic Geometry

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  1. Topics in Algebraic Geometry By: Abraham Taicher Thomas Murphy With: Amanda Knecht (grad) Brendan Hassett (prof)

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

  3. 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

  4. Example: singularity • F=xy-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.

  5. Example: AnotherSingularity • F=x2y+xy2-x4-y4 • =y2+2xy-4x3 • =x2+2xy-4y3 • Again p=(0,0) is the only point that satisfies all three conditions. • This singularity is called an ordinary triple point.

  6. 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(x2y+xy2-x4-y4)=3 • Notice that μ(f)<2  f is nonsingular at the origin.

  7. 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.

  8. Resolving Singularities: Blowing Up

  9. Example: Blowup Final graph

  10. Differential Forms • Differential forms dxΛdy follow two rules: • dxΛdx=0 • d(uy)=yduΛudy • We can trace the progression of differential forms through the process of blowing up.

  11. Example: Differential Forms • F=x2-y3 • Blowup 1: • Substitute x=uy • Blowup 2: • Substitute y=au • Blowup 3: • Substitute a=bu dxΛdy => (ydu+udy)Λdy = yduΛdy => auduΛ(adu+uda) =u2aduΛda => u2buduΛ(bdu+udb) =u4bdu Λdb

  12. Properties: Log Canonical Threshold • To calculate the log canonical threshold α, 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.

  13. The Problem • What is the Log Canonical Threshold for • f=xp-yq ? • Methods: • Combinatorics involving continued fractions. • The geometry of blowups and the adjunction formula.

  14. IT’S OVER!

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