Blowing Things Up, Algebraic Geometry Style

1. Blowing Things Up, Algebraic Geometry Style Clendenen, Jamshidi, Kirton, Lax

2. Outline Introduction Plane Curve Singularities The Process of Blowing Up Log Canonical Threshold of a Function Types of Equivalences Current and Future Projects

3. Introduction What is Algebraic Geometry? What kinds of functions are we using? What field are we working over? What is our goal for this project? Study of behavior of polynomial equations Polynomials Complex numbers Understand more about singularity resolutions

4. Isolated Singularities Non-singular graph of y2-x(x+1)(x+2)=0 What is a Plane Curve? What is a Singularity? When is a singularity isolated? f(x,y)=0 Point where f=dxf=dyf=0 Only singularity in a small neighborhood

5. y2-x2=0 Example of a Node y x

6. y x y2-x3=0 Example of a Cusp

7. Example of an Ordinary Triple Point

8. Second Example of a Node y2-x2-x3=0

9. The Process of Blowing Up

10. y C C: y2-x2-x3=0 x C’ Let y= sx s2x2-x2-x3=0 x2(s2-1-x)=0 E1 C’ A Simple Example

11. An illustration of this process:

12. C y x A More Complicated Example C: y2-x5=0

13. C y Let y=sx s2x2-x5=0 x2(s2-x3)=0 Let x=at a4t4(t2-at)=0 a4t5(t-a)=0 x E1 E3 C’’’ C’ E2 C’ C’’’ E1 E3 Let s=tx x2(t2x2-x3)=0 x4(t2-x)=0 C’’ E1 C’’’’ Let a=bt b4t9(t-bt)=0 b4t10(1-b)=0 E2 E2 E4 E1 C’’ C’’’’ Step 1: Step 2: C: y2-x5=0 Step 3: Step 4:

14. Ki+1 Ci Log Canonical Threshold The LCT of a singularity is a value that is associated with its blow up process At every step in the blow up process, we can calculate two “multiplicities” of the function: the multiplicity of the exceptional divisor (Ci), and the canonical divisor multiplicity (Ki) The LCT is given by the infimum (minimum) of:

15. Let x=at a4t4(t2-at)=0 a4t5(t-a)=0 Let y=sx s2x2-x5=0 x2(s2-x3)=0 C’’’ C’ Let s=tx x2(t2x2-x3)=0 x4(t2-x)=0 Let a=bt b4t9(t-bt)=0 b4t10(1-b)=0 C’’ C’’’’ C: y2-x5=0 Step 2: C2=4 C1=2 Step 1: x2dtdx K2=2 xdsdx K1=1 a2=(2+1)/4=3/4 a1=(1+1)/2=1 Step 4: Step 3: C4=10 C3=5 b2t6dbdt K4=6 a2t3dadt K3=3 a4=(6+1)/10=7/10 LCT=7/10 a3=(3+1)/5=4/5

16. 1 1 p q Theorem LCT(xp-yq,0) = + From previous example, LCT(x5-y2,0)= 1/5 + 1/2 = 7/10 First proved by Igusa

17. Theorem: One Use of the LCT Look at f(z,w)=0. Assume (0,0) is an isolated singularity. Consider the following integral: Example:

18. Types of Equivalences

19. Analytic Equivalence Two functions are analytically equivalent if there exists an invertible power series mapping that takes one function to the other (at a given point).

20. y Zooming in x Example: y2-x2-x3=0 and y2-x2=0 are analytically equivalent at the origin.

21. Topological Equivalence Two functions are topologically equivalent at a singularity if the intersections in the blow-up drawings are the same (but the equations are not necessarily the same)

22. Example: y12-x8=0 (y3+x2)(y3-x2)(y3-2x2)(y3-3x2)=0 and are not analytically equivalent, but after blowing up, the graphs of both equations are: Thus, these functions are topologically equivalent at the origin.

23. Analytic Equivalence Topological Equivalence Topological Equivalence Equivalent LCT

24. Current and Future Projects

25. We want to know for what values of f + g = 0 and f = 0 have certain equivalences. It is currently not known for what constraints of will allow LCT(f+g) = LCT(f). Given a polynomial f, we consider deformations of the form for “small” i and j (depending on f).

27. Special Thanks To: Dr. Hassett Dr. Varilly Zhiyuan Li Shuijing Li Ben Waters National Science Foundation

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