Resolving Singularities

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Resolving Singularities

• One of the Wonderful Topics in Algebraic Geometry

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Group Members
David Eng Will Rice, 2008 Ian Feldman Sid Rich, 2009 Ian Feldman Sid Rich, 2009 Robbie Fraleigh Will Rice, 2009
Itamar Gal SUNY Stony Brook, 2007 Daniel Glasscock Brown, 2009 Daniel Glasscock Brown, 2009 Taylor Goodhart Sid Rich, 2009
Aaron Hallquist Will Rice, 2009 Dugan Hammock UT-Austin, 2007 Dugan Hammock UT-Austin, 2007 Patrocinio Rivera Sid Rich, 2009
Justin Skowera Baker, 2007 Justin Skowera Baker, 2007 Justin Skowera Baker, 2007 Justin Skowera Baker, 2007
Amanda Knecht Mathematics Graduate Student, Rice University Amanda Knecht Mathematics Graduate Student, Rice University Matthew Simpson Mathematics Graduate Student, Rice University Matthew Simpson Mathematics Graduate Student, Rice University
Dr. Brendan Hassett Professor of Mathematics, Rice University Dr. Brendan Hassett Professor of Mathematics, Rice University Dr. Brendan Hassett Professor of Mathematics, Rice University Dr. Brendan Hassett Professor of Mathematics, Rice University
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The Goal

• To find out how we can deform a polynomial
  without changing certain key characteristics
• The characteristic we care about is the Log
  Canonical Threshold

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What is Algebraic Geometry?

• Algebraic Geometry is the study of the zero-sets
  of polynomial equations
• An algebraic curve is defined by a polynomial
  equation in two variables f y2 - x2 - x3 0

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What are Singularities?

• A singularity is a point where the curve is no
  longer smooth or intersects itself
• Specifically, a singularity occurs when the
  following is satisfied

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A Singularity
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Reasons to Study Singularities

• Singularities help us better understand certain
  curves
• Computers dont like to graph singularities, so
  alternative methods are needed

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Matlab Fails

• At the start, the graph looks OK
• As we zoom in, though, we begin to see a problem
• The Matlab algorithm cannot graph at a singular
  point

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How Do We Fix This?

• The blow-up technique stretches out the curve
  so it becomes smooth
• We create a third dimension based on the slope of
  the singular curve

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The Theory

• Singular curves can be plotted as
  higher-dimensional smooth curves
• You get the singular curve by looking at the
  shadow of the smooth curve

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Blow-Ups

• The blow-up process gives us new information
  about our singular curve
• In the case of y2 - x2 - x3 0 it takes only one
  blow-up to resolve the singularity and get a
  smooth curve
• Sometimes it takes many blow-ups before we end up
  with a smooth curve in higher-dimensional space

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Example Blow-Ups

• This is an example of the blow-up process
• The function we will use is a sextic plane curve
  sometimes called The Butterfly Curve

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Example Blow-Ups

• We make a substitution for x based on the
  functions slope
• We plot the result to see if it is smooth
• Theres a singularity at (0,0)

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Example Blow-Ups

• We do another substitution to get rid of this new
  singularity
• Again, we get a new singular curve, so we repeat
  the process once more

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Example Blow-Ups

• We again substitute for t
• Our plot, though unusual, is non-singular
• This means our singularity is resolved

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Example Blow-Ups

• We can now calculate the Log Canonical Threshold
  for this singularity
• It uses information (the As and Es) gained during
  the blow-up process

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Curve Resolver

• To make our lives easier, Taylor Goodhart wrote a
  program called Curve Resolver
• The program automates the blow-up process
• The program uses Java along with Mathematica to
  perform the necessary calculations

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Curve Resolver
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What Were Studying

• Curve Resolver also calculates some properties
  (called invariants) used to classify curves
• The invariant we care about is called the Log
  Canonical Threshold, which measures the
  simplicity of a singularity

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Log Canonical Thresholds
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Log Canonical Thresholds
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Log Canonical Thresholds

• We use information from the blow-up process to
  calculate the Log Canonical Threshold
• The Log Canonical Threshold can also be
  calculated using the following formula

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Our Research

• We want to find ways to keep the Log Canonical
  Threshold constant while deforming a curve
• We deform by adding a monomial

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Newton Polygon

• We can use a geometric object called a Newton
  Polygon to find the Log Canonical Threshold

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Example y6 x2y x4y5 x5

• We start with the y6 term
• The x power is 0 while the y power is 6
• It is plotted at (0,6)

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Example y6 x2y x4y5 x5

• The process continues for the other points
• x2y goes to (2,1)

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Example y6 x2y x4y5 x5

• The process continues for the other points
• x2y goes to (2,1)
• x4y5 goes to (4,5)

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Example y6 x2y x4y5 x5

• The process continues for the other points
• x2y goes to (2,1)
• x4y5 goes to (4,5)
• x5 goes to (5,0)

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Example y6 x2y x4y5 x5

• We now add the positive quadrant to all the
  points
• The Newton Polygon is defined to be the convex
  hull of the union of these areas

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Example y6 x2y x4y5 x5

• We now add the positive quadrant to all the
  points
• The Newton Polygon is defined to be the convex
  hull of the union of these areas
• Thusly.

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Example y6 x2y x4y5 x5

• Finally we draw the y x line
• It intersects the polygon at (12/7,12/7)
• 7/12 is an upper bound for the Log Canonical
  Threshold

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Example y6 x2y x4y5 x5

• In this case, the Log Canonical Threshold
  actually is 7/12
• We have preliminary results which detail when our
  bound gives the actual threshold

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Future Expansion

• We want to develop general forms for all curves
  with certain Log Canonical Thresholds
• Understanding how we can deform a curve and keep
  other invariants constant