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 \(H_4(X;\mathbf{Z})\).

An adjunction

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 \(\beta\cdot E=1\).