Resolving Singularities One of the Wonderful Topics in Algebraic Geometry

Group Members David Eng Will Rice, 2008 Ian Feldman Sid Rich, 2009 Robbie Fraleigh Will Rice, 2009 Itamar Gal SUNY Stony Brook, 2007 Daniel Glasscock Brown, 2009 Taylor Goodhart Sid Rich, 2009 Aaron Hallquist Will Rice, 2009 Dugan Hammock UT-Austin, 2007 Patrocinio Rivera Sid Rich, 2009 Justin Skowera Baker, 2007 Amanda Knecht Mathematics Graduate Student, Rice University Matthew Simpson Mathematics Graduate Student, Rice University Dr. Brendan Hassett Professor 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 = y 2 - x 2 - x 3 = 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:

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 y 2 - x 2 - x 3 = 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 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 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 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

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

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

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:

Our Research 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

Example: y 6 + x 2 y + x 4 y 5 + x 5 We start with the y 6 term The x power is 0 while the y power is 6 It is plotted at (0,6)

Example: y 6 + x 2 y + x 4 y 5 + x 5 The process continues for the other points x 2 y goes to (2,1)

Example: y 6 + x 2 y + x 4 y 5 + x 5 The process continues for the other points x 2 y goes to (2,1) x 4 y 5 goes to (4,5)

Example: y 6 + x 2 y + x 4 y 5 + x 5 The process continues for the other points x 2 y goes to (2,1) x 4 y 5 goes to (4,5) x 5 goes to (5,0)

Example: y 6 + x 2 y + x 4 y 5 + x 5 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: y 6 + x 2 y + x 4 y 5 + x 5 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.

Example: y 6 + x 2 y + x 4 y 5 + x 5 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

Example: y 6 + x 2 y + x 4 y 5 + x 5 In this case, the Log Canonical Threshold actually is 7 / 12 We have preliminary results which detail when our bound gives the actual threshold

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