Vigre 2009: algebraic geometry

Example 2.4 Consider $f(x,y)=y^2-x^3$ . As shown in example [link] , $f=0$ has a singularity at the origin.

First Blow-up

In this equation, x ² is the exceptional divisor while $s^2-x$ is what is called the proper transform . Note that the proper transform is tangent to the exceptional divisor.

Second Blow-up

In this equation, t ² and s ³ are exceptional divisors while $s-t$ is the proper transform. Note that the proper transform intersects two exceptional divisors at one point.

Third Blow-up

In this equation, r ² and s ⁶ are exceptional divisors while $1-r$ is the proper transform. Note that the proper transform is now resolved.

Definition 2.5 The multiplicity of the exceptional divisor is the highest degree of the factored variables of the equation provided after a blow-up substitution. We denote the multiplicity of the $i^{th}$ exceptional divisor as C _i .

Example 2.6 Using the blow-up sequence in Example [link] , we can see that $C_1=2$ , $C_2=3$ , and $C_3=6$ .

Definition 2.7 When blowing-up, we track the result of the series of substitutions leading up to and including that blow-up on the original differential form (in our examples $dxdy$ ). Note that the change of variables formula will always contribute an additional power to the higher degree variable of the differential form. The multiplicity of the canonical divisor is the highest degree of the factored variables of the differential form. We denote the multiplicity of the $i^{th}$ canonical divisor as K _i .

Example 2.8 Using the blow-up sequence in Example [link] , we start with $dxdy$ . After the first blow-up, the change of variables gives us the differential form $xdsdx$ . After the second blow-up, the differential form is $t{s}^{2}dtds$ . After the third blow-up, the differential form is $r{s}^{4}drds$ . We can see that $K_1=1$ , $K_2=2$ , and $K_3=4$ .

Definition 2.9 Given a normal crossings blow-up sequence with multiplicities of the canonical divisor and the exceptional divisor known, we calculate ${\alpha }_i=\frac{K_i+1}{C_i}$ . The log canonical threshold of the function is the infimum of the set of α _i for all i . We denote the log canonical threshold of a function f at a singularity p as $LCT(f,p)$ .

Example 2.10 Using the blow-up sequence in Example [link] , we can see that ${\alpha }_1=1$ , ${\alpha }_2=1$ , and ${\alpha }_3=\frac{5}{6}$ . Thus, $LCT(f,0)=\frac{5}{6}$ .

Continued fractions

The proof of the Theorem [link] relies heavily on continued fractions. In this section we recall some basic facts about continued fractions and prove some Lemmas about them relevant to the theorem. Consider the rational number $\frac{q}{p}$ . First, using the Euclidean algorithm, we write:

The continued fraction of $\frac{q}{p}$ is

This can be abbreviated with the standard notation for continued fractions: $\frac{q}{p}=[r_1,r_2,r_3,r_4,...]$ . Since $\frac{q}{p}$ is a rational number, we know that the continued fraction expression will eventually terminate, meaning that there exists a smallest positive integer δ such that $n_{\delta }=0$ .

We call $[r_1,r_2,...,r_v]$ the $v^{th}$ convergent of $\frac{q}{p}$ . Note that the ${\delta }^{th}$ convergent equals $\frac{q}{p}$ . We will then rearrange the variables of the continued fraction convergents so that they are in the form of a simple fraction. For example, we will rearrange the second convergent of $\frac{q}{p}$ to be ${\displaystyle \frac{r_1{r}_2+1}{r_2}}$ ; the third convergent will become $r_1+{\displaystyle \frac{r_3}{r_2{r}_3+1}}={\displaystyle \frac{r_1{r}_2{r}_3+r_2+r_3}{r_2{r}_3+1}}$ ; etc. Once the $v^{th}$ convergent is in this final form, we will denote the numerator of this expression h _v and the denominator of this expression k _v . Denote $\frac{h_v}{k_v}$ by A _v . Notice that $\frac{q}{p}=A_{\delta }$ , since the ${\delta }^{th}$ convergent is the entire continued fraction.