de Jong: Rationally Connected Varieties

Deformation theory

Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $C \hookrightarrow X$ be a one-dimensional closed subscheme. We have $\mathscr{O}_X \supset \mathscr{I}_C$, the ideal sheaf of $C$, and we assume that $C$ is a local complete intersection, or what is equivalent, $\mathscr{I}_C/\mathscr{I}_C^2$ is a locally free sheaf of $\mathscr{O}_C$-modules of rank $\dim X-1$. For example, this holds if $X$ is a nodal curve.

Definition. The normal bundle of the curve $C$ in $X$ is

$$\mathscr{N}_C X=\mathscr{H}om_{\mathscr{O}_C}(\mathscr{I}_C/\mathscr{I}_C^2,\mathscr{O}_C).$$

We have that

$$\widetilde{\mathscr{O}}_{{\mathrm{Hilb}}_X,[C]}=\mathbb{C}[[t_1,\dots,t_d]]/(f_1,\dots,f_r),$$

where $d=h^0(C,\mathscr{N}_C X)$, and $r \leq h^1(C,\mathscr{N}_C(X))$. We say that $H^0$ is the deformation space, and $H^1$ are the obstructions to deformation. In particular, the dimension the Zariski tangent space of the Hilbert scheme at $[C]$ has $\mathscr{T}_{[C]} {\mathrm{Hilb}}_X=H^0(C,\mathscr{N}_C X)$, and it has dimension at least $\chi(\mathscr{N}_C X)$, the Euler characteristic.

Example. In the case where $[f:C \to \mathbb{P}^1] \in \overline{\mathcal{M}_g}(\mathbb{P}^1,d)$, with $C$ is a smooth genus $g$ curve, and $f$ having at worst simple branchings, then there are no obstructions to deformation and

$$\mathscr{T}_{[f]} \overline{\mathcal{M}_g}(\mathbb{P}^1,d)=\bigoplus_{P\in{\mathrm{Ram}}(f)} \mathscr{T}_{\mathbb{P}^1}\vert _{f(P)}.$$

This is a canonical way of understanding how to move branch points on maps to $\mathbb{P}^1$.

A map of moduli spaces

Let $f:X \to \mathbb{P}^1$ be a nonconstant morphism with $X$ a nonsingular projective variety over $\mathbb{C}$, and let $C \subset X$ be a closed subscheme which is a smooth curve of genus $g$, such that the ramification of $f\vert _C$ is simple. In a (formal) neighborhood of $[C \subset X]$, the spaces $\overline{\mathcal{M}_g}(X,[C])$ and the ${\mathrm{Hilb}}_X$ are the same. The map

$$\overline{\mathcal{M}_g}(X,[C]) \to \overline{\mathcal{M}_g}(\mathbb{P}^1,d)$$

where $d$ is the degree of $f$ on $C$, induces on tangent spaces

$$H^0(C,\mathscr{N}_C X) \to \textstyle{\bigoplus}_{P\in{\mathrm{Ram}}(f\vert _C)} \mathscr{T}_{\mathbb{P}^1}\vert _{f(P)}$$

induced by the right vertical arrow in the diagram

$$\xymatrix{ 0 \ar[r] & \mathscr{T}_C \ar@{=}[d] \ar[r] & \mathscr{... ...rt _C \ar[r] & (f^*\mathscr{T}_{\mathbb P^1}\vert C)/\mathscr{T}_C \ar[r] & 0 }$$

If $C$ is contained in the smooth locus of $f$, then the middle vertical map $\mathscr{T}_X\vert _C \to f^*\mathscr{T}_{\mathbb{P}^1}\vert _C$ is surjective, hence also the right vertical map $\mathscr{N}_C X \to (f^*\mathscr{T}_{\mathbb{P}^1}\vert C)/\mathscr{T}_C$ is also surjective as maps of sheaves. This will not give a surjection on global sections, however one has:

Corollary. If $C$ is contained in the smooth locus of $f$, and $\mathscr{N}_C X$ is ``sufficiently positive'', then the morphism

$$\overline{\mathcal{M}_g}(X,[C]) \to \overline{\mathcal{M}_g}(\mathbb{P}^1,d)$$

analytically locally around $[C \subset X]$ is a projection $\mathbb{C}^{a+b} \to \mathbb{C}^b$.

One argues that the Hilbert scheme is smooth at the point $[C \subset X]$ since one can twist by a small number of points and keep that the $H^1$ vanishes. In particular, the corollary implies that the morphism is surjective.

We are now ready to prove:

Theorem. [G, Harris, Starr] If $k=\overline{k}$, ${\mathrm{char}}k=0$, then any rationally connected variety over $k(C)$ for $C$ a curve has a rational point.

Proof. Assume that $(*)$: all fibres of $X \to \mathbb{P}^1$ are reduced.

Step $1$. Take a general complete intersection $C \subset X$; it will be smooth, irreducible, of say genus $g$ and degree $d$. The condition $(*)$ implies that $C$ is in the smooth locus and $C \to \mathbb{P}^1$ (by Bertini) has at worst simple branching.

Step $2$. Choose a large integer $N$ and choose general points $c_1,\dots,c_N \in C$, and rational curves $C_i \subset X$ such that:

1. $C \cap C_i =\{c_i\}$;
2. $C_i \subset f^{-1}(f(C_i))$;
3. $\mathscr{N}_{C_i} X_{f(c_i)}$ is very ample;
4. $\mathscr{T}_{c_i} C_i \subset \mathscr{T}_{c_i} X_{f(c_i)}$ is in general position.

Now let $C^{\text{new}}=C \cup C_1 \cup \dots \cup C_N$. The basic property is that $\mathscr{N}_{C^{\text{new}}} X\vert _{C_i} \supset \mathscr{N}_{C_i} X$. Moreover, $\mathscr{N}_{C^{\text{new}}} X\vert _C \supset \mathscr{N}_C X$ with colength $N$ and assumption (iv) gives that this is ``general''. This gives that the sheaf $\mathscr{N}_{C^{\text{new}}} X$ on $C^{\text{new}}$ is sufficiently positive.

Now deform this curve to a simply branched curve, and this gives the result; conclude by the corollary. $\qedsymbol$

Multiple fibres

We must deal with the case when $(*)$ fails. Suppose we have a family of varieties $X \to \mathbb{P}^1$ with fibres at $t_1,\dots,t_r$ irreducible of multiplicity $m_1,\dots,m_r$. Since the curve must intersect these fibres transversally, this must be preserved in any deformation, meaning that the ramification index at $t_i$ will be divisible by $m_i$.

In this case, the problem is: $\overline{\mathcal{M}_g}(X,\beta) \to \overline{\mathcal{M}_g}(\mathbb{P}^1,d)$ cannot dominate. Instead, we consider consider the subset $Z_{g,d}^{(t_i,m_i)} \subset \overline{\mathcal{M}_g}(\mathbb{P}^1,d)$ consisting of stable maps $f:C \to \mathbb{P}^1$, $C$ of genus $g$, $f$ of degree $d$, such that all ramification indices above $t_i$ are equal to $m_i$.

Now we have the additional problems: Which reducible curves are in $Z_{g,d}^{(t_i,m_i)}$? And perhaps $\dim Z$ is too small? To resolve both problems, enlarge the genus $g$ (but not $d$) by adding loops to $C$: join two points with a good rational curve. This allows you to break off a component even in this case.

Conclusion

This is work with Jason Starr. What will guarantee the existence of a rational point on a variety over a function field in two variables? Is there a geometric condition which would be like rational connectedness in this case? This is too much to hope for, there are many surfaces $S$ with a nontrivial Brauer-Severi variety $X \to S$. Maybe there are geometric restrictions on the fibres $X \to S$ such that one obtains a rational section.

A good guess for this condition: demand that certain moduli spaces of rational curves on the fibers are themselves rationally connected. For example, Starr and Harris proved that for hypersurfaces $X$ of degree $d$ in $\mathbb{P}^n$ with $d^2+d+2 \leq n$, the moduli spaces of rational curves of fixed degree on $X$ are themselves rationally connected.

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