Curves on surfaces of mixed characteristic

Abstract

The classification of foliated surfaces (McQuillan in Pure Appl Math Q 4(3):877–1012, 2008) is applied to the study of curves on surfaces with big co-tangent bundle and varying moduli, be it purely in characteristic zero, or, more generally when the characteristic is mixed. Almost everything that one might naively imagine is true, but with one critical exception: rational curves on bi-disc quotients which aren’t quotients of products of curves are Zariski dense in mixed characteristic. The logical repercussions in characteristic zero of this exception are not negligible.

Appendix: Baum–Bott theory with values in the ground field

Let X / S be an algebraic, resp. complex, Deligne–Mumford champ over an algebraic, resp. complex, space. The Deligne–Mumford condition ensures that ${\Omega }^1_{X/S}$ is well defined, and whence for every $n\in \mathbb {N}$ there is, on identifying $\mathrm {H}^1$ with Čech co-cycles, an Atiyah class,

(101)

whose symmetric functions define for each $q\in \mathbb {N}$ Chern-classes of “Hodge type”, i.e.

$$\begin{aligned} {\mathrm {c}}_q:\mathrm {H}^1(X, {\mathrm {GL}}_n)\rightarrow \mathrm {H}^q(X, {\Omega }^q_{X/S}). \end{aligned}$$

(102)

As such the Atiyah class of a vector bundle, E, vanishes iff there is a connection

and such vanishing implies, a fortiori, vanishing of the Chern classes of (102). Naturally, therefore, the Atiyah class is usually viewed as an obstruction, and, rather obviously, if one only asks that E admits a connection along certain directions then this obstruction is smaller, i.e. for any derivation with values in an -module

Fact A.1

The obstruction that a vector bundle E admits a connection along D, i.e. a map

satisfying the Leibniz rule, , lies in

(103)

Proof

By definition, D corresponds to a map , so that the obstruction (103) is just the restriction of the Atiyah class (101). $\square$

A generally valid example of such a bundle would be

Example A.2

If C is a family of invariant effective Cartier divisor, then by definition

so we get (rather tautological) connections for , and ${\mathscr {O}}_X(C)$.

Consequently, irrespective of any integrability about the field of directions afforded by D

Corollary A.3

If $f:Y\rightarrow X$ is a map such that we have a factorisation

$$\begin{aligned} f^*{\Omega }^1_{X/S} \rightarrow f^*{\mathscr {M}}\rightarrow {\Omega }^1_{Y/S} \end{aligned}$$

then for any bundle E admitting a connection along D the Hodge Chern classes

vanish.

Now the key point is that in the presence of integrability there is at least one rather interesting bundle admitting a connection along the leaves, i.e.

Fact A.4

Suppose the sheaf of derivations, $T_{\mathscr {F}}$, vanishing along the kernel, ${\Omega }^1_{X/{\mathscr {F}}}$, of ${\Omega }^1_{X/S}\rightarrow {\mathscr {M}}$ is closed under bracket and $U\hookrightarrow X$ is the locus where ${\Omega }^1_{X/{\mathscr {F}}}\hookrightarrow {\Omega }^1_{X/S}$ defines a sub-bundle of ${\Omega }^1_{X/S}$, then ${\Omega }^1_{X/{\mathscr {F}}}|_U$ admits a connection along ${\Omega }^1_{X/S}|_ U \rightarrow {\Omega }^1_{\mathscr {F}}:=\mathrm {Hom}_{{\mathscr {O}}_X}(T_{\mathscr {F}}, {\mathscr {O}}_X)$.

Proof

Differentiation affords a map

(104)

so by hypothesis the former map in (104) factors through the kernel of the latter, which over the locus, U, is naturally the image of

and the quotient of this by the sub-image of is, over, U: . $\square$

Typically one applies this in the form

Corollary A.5

Suppose ${\Omega }^1_{X/{\mathscr {F}}}, {\Omega }^1_{\mathscr {F}}$ are bundles; U is of co-dimension 2; and X is $S_2$ then ${\Omega }^1_{X/{\mathscr {F}}}$ admits a connection,

(105)

Nevertheless,

Warning A.6

It does not follow from A.3 and A.5 that for $f:Y\rightarrow X$ factoring through a ${\mathscr {F}}$ invariant sub-variety that the Atiyah class, or even the Chern classes, whether of $f^* {\Omega }^1_{X/{\mathscr {F}}}$, or of A.2, vanishes. Indeed this only follows if (105) factors, around f, through

otherwise there is a residue/obstruction in

The resulting residues admit, [2] or [17, 1.3] for a characteristic free version, various formulae, of which the most relevant is:

Example A.7

Let $(X,{\mathscr {F}})$ be a foliated smooth surface with log-canonical singular locus Z over a field k for which every singularity is a reduced k-point and E an invariant simple normal crossing divisor such that at any singularity there are two branches of E then ${\Omega }^1_{X/F}(E)$ admits a connection without residues, i.e.

(106)

In particular, for p the characteristic of k and $f:{\Sigma }\rightarrow X$ any invariant curve

(107)

Proof

The in particular follows from the general residue formula [17, 1.3.1] of which (106) is a minor variation. $\square$