Resolving Singularities

One of the Wonderful Topics in Algebraic Geometry

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

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

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

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

A Singularity

Reasons to Study Singularities

Singularities help us better understand certain curves

Computers don’t like to graph singularities, so alternative methods are needed

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

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

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

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

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”

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)

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

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

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

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

Curve Resolver

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

Log Canonical Thresholds

1

1

22

LCT

xy

6

5

32

LCT

xy

Log Canonical Thresholds

4

3

42

LCT

xy

2

1

2 2283

LCT

yxxy

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

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

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

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

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

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

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

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

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

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

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

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