1 / 33

Resolving Singularities

Resolving Singularities. One of the Wonderful Topics in Algebraic Geometry. Group Members. 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?.

nanji
Download Presentation

Resolving Singularities

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Resolving Singularities One of the Wonderful Topics in Algebraic Geometry

  2. Group Members

  3. 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

  4. 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

  5. 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:

  6. A Singularity

  7. Reasons to Study Singularities • Singularities help us better understand certain curves • Computers don’t like to graph singularities, so alternative methods are needed

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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”

  13. 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)

  14. 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

  15. Example: Blow-Ups • We again substitute for t • Our plot, though unusual, is non-singular • This means our singularity is resolved

  16. 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

  17. 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

  18. Curve Resolver

  19. 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

  20. Log Canonical Thresholds

  21. Log Canonical Thresholds

  22. 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:

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

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

  25. 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)

  26. Example: y6 + x2y + x4y5 + x5 • The process continues for the other points • x2y goes to (2,1)

  27. Example: y6 + x2y +x4y5 + x5 • The process continues for the other points • x2y goes to (2,1) • x4y5 goes to (4,5)

  28. 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)

  29. 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

  30. 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.

  31. 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

  32. 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

  33. 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

More Related