# Discrepancies and intersections in orbifolds

## 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 \(k_i\).

*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 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 \(f\).