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Resolving Singularities
One of the Wonderful Topics in Algebraic Geometry
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Group MembersDavid Eng
Will Rice, 2008
Ian FeldmanSid Rich, 2009
Robbie FraleighWill Rice, 2009
Itamar GalSUNY Stony Brook, 2007
Daniel GlasscockBrown, 2009
Taylor GoodhartSid Rich, 2009
Aaron HallquistWill Rice, 2009
Dugan HammockUT-Austin, 2007
Patrocinio RiveraSid Rich, 2009
Justin SkoweraBaker, 2007
Amanda KnechtMathematics Graduate Student, Rice
University
Matthew SimpsonMathematics Graduate Student, Rice
University
Dr. Brendan HassettProfessor 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:
0,0,at y Singularit
0232
0
2322
yx
yxxxxy
y
f
x
ff
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A Singularity
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Reasons to Study Singularities
Singularities help us better understand certain curves
Computers don’t 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
CurveButterfly The
xxy 0626 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
2 1
0
0
0
11
461
21
42
661
221
6
1
626
EA
yttyy
ytyty
ytx
xxy
We make a
substitution for x based on the function’s slope
We plot the result to see if it is smooth
There’s a singularity at (0,0)
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Example: Blow-Ups
4 2
0)(
0
0
22
862
22
24
1062
222
42
21
461
21
42
EA
yttyy
ytytyy
ytt
yttyy
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
6 3
0)1(
0
0)(
33
1263
23
6
1463
223
24
32
862
22
24
EA
ytty
ytytyy
ytt
yttyy
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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
3
2
3
2,
4
3,1min
1min
i
i
E
ALCT
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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 We’re 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
1
1
22
LCT
xy
6
5
32
LCT
xy
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Log Canonical Thresholds
4
3
42
LCT
xy
2
1
2 2283
LCT
yxxy
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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:
1,0 converges, 1
inf 2 f
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Our Research
3
2
33
LCT
xy
3
2
233
LCT
yxxy
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
0
1
2
3
4
5
6
7
0 2 4 6
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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)
0
1
2
3
4
5
6
7
0 2 4 6
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Example: y6 + x2y + x4y5 + x5
The process continues for the other points
x2y goes to (2,1)
0
1
2
3
4
5
6
7
0 2 4 6
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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)
0
1
2
3
4
5
6
7
0 2 4 6
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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)
0
1
2
3
4
5
6
7
0 2 4 6
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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.
0
1
2
3
4
5
6
7
0 2 4 6
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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
0
1
2
3
4
5
6
7
0 2 4 6
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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
0
1
2
3
4
5
6
7
0 2 4 6
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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