Understanding Singularities and Log Canonical Thresholds in Algebraic Geometry

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