# Discrepancies and intersections in orbifolds

[2018-06-04 Mon]

This is the second of my posts covering background material on cyclic quotient singularities of surfaces (the first one is here). In this post, I will discuss discrepancies and intersection numbers in surfaces with singularities, and explain how to compute them.

## Discrepancies

### Definition

Consider a cyclic quotient surface singularity $$X$$ of type $$\frac{1}{P}(1,Q)$$ and let $$\pi\colon\tilde{X}\to X$$ be its minimal resolution, with exceptional locus $$E_1\cup\ldots\cup E_r$$. Recall from the last post that the exceptional locus is a chain of embedded spheres with self-intersections $$E_i^2=-b_i$$ where the continued fraction of $$P/Q$$ is $$[b_1,\ldots,b_r]$$. Since $$\tilde{X}$$ is a complex manifold, its tangent bundle has a first Chern class $$c_1(\tilde{X})\in H^2(\tilde{X};\mathbf{Z})$$; we will work instead with the canonical class $$K_{\tilde{X}}=-c_1(\tilde{X})$$. We would like to express $$K_{\tilde{X}}$$ in terms of the Poincaré duals of the submanifolds $$E_i$$. The problem is that $$\tilde{X}$$ is not compact, so Poincaré duality fails and we only have Alexander-Lefschetz duality $$H^2(\tilde{X})\cong H_2(\tilde{X},\partial\tilde{X})$$. Certainly the classes $$E_i$$ make sense in $$H_2(\tilde{X},\partial\tilde{X};\mathbf{Z})$$, but they are no longer generators: they span the image of the map $$H_2(\tilde{X};\mathbf{Z})\to H_2(\tilde{X},\partial\tilde{X};\mathbf{Z})$$, whose cokernel is $$H_1(\partial\tilde{X};\mathbf{Z})$$ by the long exact sequence of the pair $$(\tilde{X},\partial\tilde{X})$$. Since the boundary of a neighbourhood of a cyclic quotient singularity of type $$\frac{1}{P}(1,Q)$$ is a lens space of type $$L(P,Q)$$, this cokernel is isomorphic to $$\mathbf{Z}/(p)$$.

If we work over $$\mathbf{Q}$$ instead then this cokernel vanishes and, consequently, we can express $$K_{\tilde{X}}$$ as a rational linear combination of the Alexander-Lefschetz duals $$[E_i]$$ to the exceptional spheres: $K_{\tilde{X}}=\sum k_i[E_i].$ The numbers $$k_i$$ are called the discrepancies of the singularity (they measure the discrepancy between $$K_{\tilde{X}}$$ and $$\pi^*K_X$$, where $$\pi^*K_X=0$$ in this case).

### Computing discrepancies

To compute the discrepancies, we note that each curve $$E_i$$ satisfies the adjunction formula: $K_{\tilde{X}}\cdot[E_i]=-E_i^2-\chi(E_i)=b_i-2,$ where $$\chi(E_i)$$ denotes the Euler characteristic of $$E_i$$ which is 2 since $$E_i$$ is a sphere. This case of the adjunction formula is just a statement that the first Chern class is additive under Whitney sum of complex vector bundles and $$T\tilde{X}|_{E_i}=\nu\tilde{X}_{E_i}\oplus TE_i$$ as complex vector bundles (where $$\nu\tilde{X}_{E_i}$$ is the normal bundle of $$E_i$$ in $$\tilde{X}$$, whose first Chern class is the self-intersection of $$E_i$$).

We have $$K_{\tilde{X}}=\sum k_jE_j$$, so $K_{\tilde{X}}\cdot[E_i]=\sum_j k_jE_j\cdot E_i,$ so this gives us $$r$$ simultaneous equations: $\sum_j k_jE_j\cdot E_i=b_i-2$ for the $$r$$ numbers $$k_r\in\mathbf{Q}$$. This is enough to determine the $$k_i$$.

Consider the $$\frac{1}{n}(1,1)$$ singularity. Since the continued fraction of $$n/1$$ is $$n$$, the minimal resolution has one exceptional curve $$E_1$$ with $$E_1^2=-n$$. Adjunction for $$E_1$$ tells us that $$k_1n=n-2$$, so $$k_1=\frac{n-2}{n}$$. In particular, we see that the minimal resolution of a $$\frac{1}{2}(1,1)$$ singularity (ordinary double point) is Calabi-Yau ($$K_{\tilde{X}}=0$$).

We can actually give a formula for the discrepancies, which I found in this paper by Hacking, Tevelev and Urzúa, and which I keep having to work out from scratch because I forget it. The formula uses partial continued fractions.

• There exist positive integers $$\alpha_1,\ldots,\alpha_r$$ such that any two consecutive $$\alpha_i$$ are coprime and $\frac{\alpha_i}{\alpha_{i-1}}=[b_{i-1},\ldots,b_1].$ In particular, $$\alpha_1=1$$, $$\alpha_{r}=\bar{Q}$$, where $$\bar{Q}$$ is the multiplicative inverse of $$Q$$ modulo $$P$$.
• There exist positive integers $$\beta_1,\ldots,\beta_r$$ such that any two consecutive $$\beta_i$$ are coprime and $\frac{\beta_i}{\beta_{i+1}}=[b_{i+1},\ldots,b_r].$ In particular, $$\beta_1=Q$$ and $$\beta_r=1$$.
• The numbers $$k_i=-1+\frac{\alpha_i+\beta_i}{P}$$ solve the simultaneous equations for the discrepancies. As a corollary, $$k_1=\frac{1-(P-Q)}{P}$$ and $$k_r=\frac{1-(P-\bar{Q})}{P}$$.
We construct the numbers $$\alpha_i$$ by induction starting with $$\alpha_2/\alpha_1=[b_1]=b_1$$, so take $$\alpha_1=1$$, $$\alpha_2=b_1$$ (which are certainly coprime). Assume that we have constructed $$\alpha_1,\ldots,\alpha_i$$. We need to solve $$\alpha_{i+1}/\alpha_i=[b_i,\ldots,b_1]$$. We have \begin{align*} [b_i,\ldots,b_1]&=b_i-\frac{1}{[b_{i-1},\ldots,b_1]}\\ &=b_i-\frac{\alpha_{i-1}}{\alpha_i}\\ &=\frac{b_i\alpha_i-\alpha_{i-1}}{\alpha_i}, \end{align*} so we may take $$\alpha_{i+1}=b_i\alpha_i-\alpha_{i-1}$$. To see that this is coprime to $$\alpha_i$$, note that $$\gcd(b_i\alpha_i-\alpha_{i-1},\alpha_i)=\gcd(\alpha_{i-1},\alpha_i)=1$$ by induction. Note that $$\frac{\alpha_{r+1}}{\alpha_r}=[b_r,\ldots,b_1]=\frac{P}{\bar{Q}}$$, which gives $$\alpha_r=\bar{Q}$$. The construction of the numbers $$\beta_i$$ is similar.

We have the identities \begin{align*} \alpha_2-\alpha_1b_1&=\alpha_1([b_1]-b_1)=0\\ \alpha_{i+1}+\alpha_{i-1}-\alpha_ib_i&=\alpha_i\left([b_i,\ldots,b_1]+\frac{1}{[b_{i-1},\ldots,b_1]}-b_i\right)\\ &=b_i-\frac{1}{[b_{i-1},\ldots,b_1]}+\frac{1}{[b_{i-1},\ldots,b_1]}-b_i\\ &=0\mbox{ for }i=2,\ldots,r-1,\\ \alpha_{r-1}-\alpha_rb_r&=\alpha_r\left(\frac{1}{[b_{r-1},\ldots,b_1]}-b_r\right)\\ &=-[b_r,\ldots,b_1]\alpha_r\\ &=\alpha_{r-1}=P, \end{align*} and \begin{align*} \beta_2-\beta_1 b_1&=-P\\ \beta_{i+1}+\beta_{i-1}-\beta_ib_i&=0\mbox{ for }i=2,\ldots,r-1,\\ \beta_{r-1}-\beta_rb_r&=0. \end{align*} Using these identities, it is easy to check that $k_i=-1+\frac{\alpha_i+\beta_i}{P}$ solves the simultaneous equations required of the discrepancies.

## Working on the singular surface

### Defining intersection numbers

It may be that we want to perform computations of first Chern classes evaluated on curves in a singular surface $$X$$, or of intersections between curves on the singular surface. If the singularities are cyclic quotient singularities (and actually much more generally: you can read all about it in Lazarsfeld's wonderful book "Positivity in Algebraic Geometry, I"), then this makes sense for the following kinds of topological reason. Let $$Z\subset X$$ be the set of singularities, let $$NZ$$ be a neighbourhood of $$Z$$ and let $$V=X\setminus NZ$$; a curve in $$X$$ defines a class in $$H_2(V,\partial V)$$. Again, by Alexander-Lefschetz duality this gives us a class in $$H^2(V)$$. By applying the Mayer-Vietoris sequence to the decomposition $$X=V\cup NZ$$, $\cdots\to H^1(\partial V)\to H^2(X)\to H^2(V)\oplus H^2(NZ)\to H^2(\partial V)\to\cdots,$ we see that $$H^2(X;\mathbf{Q})\cong H^2(V;\mathbf{Q})$$, since $$H^2(\partial V;\mathbf{Q})=H^2(NZ;\mathbf{Q})=0$$ (since $$\partial V$$ is a union of lens spaces with finite $$H_1$$ and $$NZ$$ is contractible). We also get that $$H_4(X;\mathbf{Z})\cong\mathbf{Z}$$; this is the kernel of the map $$H_3(\partial V;\mathbf{Z})\to H_3(V;\mathbf{Z})$$ induced by inclusion, and $$V$$ itself provides a nullhomology of the sum of the components of $$\partial V$$, so this sum vanishes.

Once we have turned a curve into a class in $$H^2(X;\mathbf{Q})$$, we can "intersect curves" by taking cup product to get a class in $$H^4(X;\mathbf{Q})$$ and pairing with a chosen generator for $$H_4(X;\mathbf{Z})$$.

We can pullback the first Chern classes of complex line bundles along continuous maps. If a divisor arises as the vanishing locus of a section of a holomorphic line bundle then we can "pullback" its divisor class by pulling back the first Chern class of this line bundle and then taking Poincaré-dual. If we work over $$\mathbf{Q}$$ then we can get away with working with divisors $$D$$ for which there is an integer $$n$$ such that $$nD$$ arises as the vanishing locus of a section of a line bundle. A variety on which every divisor has this property is called $$\mathbf{Q}$$-factorial, so we can define pullback of $$\mathbf{Q}$$-divisors on a $$\mathbf{Q}$$-factorial variety. We write $$f^*D$$ for the pullback of $$D$$ along a holomorphic map $$f$$.

We can also pushforward divisors along holomorphic maps just by taking the image (and keeping track of multiplicities). We write $$f_*$$ for this pushforward map.

There is an adjunction between these functors which says $f_*(f^*\alpha\cdot\beta)=f_*\alpha\cdot\beta,$ for divisor classes $$\alpha$$ on $$Y$$ and $$\beta$$ of $$X$$ where $$f\colon X\to Y$$ is a holomorphic map and the dot indicates algebraic intersection.

If $$X=\tilde{Y}$$ is the minimal resolution then we can use this formula to convert intersection number computations in $$Y$$ into intersection number computations in the minimal resolution (which are easier because it's nonsingular). For example, suppose we have two curves $$\beta_1$$ and $$\beta_2$$ on $$\tilde{Y}$$; we can compute the intersection number $f_*\beta_1\cdot f_*\beta_2$ on $$\tilde{Y}$$, as it is equal to $f_*(f^*f_*\beta_1\cdot \beta_2).$ If these divisors are curves on a surface with isolated cyclic quotient singularities then we can usually ignore the outer $$f_*$$ (this is just pushing forward points along a map which is an isomorphism away from the exceptional locus) and the key problem is to compute $$f^*f_*\beta_1$$. We have $f^*f_*\beta=\beta+\sum_jc_jE_i,$ for some numbers $$c_j$$; in other words, the discrepancy between $$\beta$$ and $$f^*f_*\beta$$ is some combination of the exceptional curves. This can be computed in much the same way as the discrepancies of the canonical class, provided one knows how $$\beta$$ intersects the curves $$E_i$$. The key observation is that $$f^*C\cdot E_i=0$$ for any divisor $$C$$, because $$f_*E_i=0$$ (the exceptional sphere is contracted by $$f$$.

Let's take a nodal quadric surface; this has a single ordinary double point. Its minimal resolution is the Hirzebruch surface $$\mathbf{F}_2$$ and the exceptional locus has one sphere $$E$$ with self-intersection $$E^2=-2$$. The Hirzebruch surface is ruled and the $$-2$$-sphere is a section. Let's take one of the rulings, $$\beta$$, and compute the self-intersection of $$f_*\beta$$ (a particular rational curve in the nodal surface passing through the node). We have $$f^*f_*\beta=\beta+cE$$ and $$0=f^*f_*\beta\cdot E=\beta\cdot E+cE^2=1-2c$$, so $$c=1/2$$. Therefore \begin{align*} (f_*\beta)^2&=f_*((f^*f_*\beta)\cdot\beta)\\ &=(\beta+E/2)\cdot\beta\\ &=1/2, \end{align*} since $$\beta^2=2$$ and $$\beta\cdot E=1$$.

Comments, corrections and contributions are very welcome; please drop me an email at j.d.evans at lancaster.ac.uk if you have something to share.

CC-BY-SA 4.0 Jonny Evans.