Rationally Connected Varieties

This is a guest post by Dylan Toh.

In this blog post, we open up the paper Families of Rationally Connected Varieties by Graber, Harris, and Starr. We explain the main result, and sketch the (frighteningly tall) tower of ideas in algebraic geometry upon which its proof perches. 

We shall be loose with terminology here. In fact, one should skip these definitions first, and only return if you find yourself wondering about technicalities. 

• A variety $X$ is a smooth projective integral scheme over $\mathbb C$, or (highly non-trivially, by Serre's GAGA) equivalently, a compact connected complex submanifold of $\mathbb C\mathbb P^N$ for some large $N$. [aside: there are many compact complex surfaces of the form $\mathbb C^2$-modulo-a-$\dim_{\mathbb R}$-$4$-lattice, which are not projective, i.e. embeddable in any $\mathbb C\mathbb P^N$.] 
• A point $p\in X$ is the usual notion of point of the complex manifold, or equivalently, a closed point of the scheme. 
• The terms immersion and embedding are also as in differential geometry. [aside: the term "closed immersion" in algebraic geometry corresponds to the term "closed embedding" in differential geometry.]
• The normal bundle of the immersion of a curve into a $\dim_{\mathbb C}$-$d$ variety is the rank-$(d-1)$ complex vector bundle of orthogonal directions. Sections of the normal bundle should correspond to ways in which one may algebraically deform the curve. It is (very) ample if for any point and direction, there is a global algebraic deformation of the curve that moves the point in that direction. [aside: there is actually a crucial distinction between ampleness and very ampleness, but we will simply ignore it.] [second aside: unlike in real differential geometry where one may always deform submanifolds, there is often much less freedom to deform complex submanifolds in complex geometry. This is to do with the rigidity of the Cauchy-Riemann equations; an analogy in real differential geometry might be to deform closed geodesics. And just as spaces may have a discrete collection of closed geodesics, a complex manifold may have no (or few) complex submanifolds.]
• For a morphism $f:X\to B$ of varieties, a general fibre is $f^{-1}(b)$ for "most points" $B$. In other words, saying that the general fibre has property $\mathcal P$ means that there is a non-empty Zariski open $U\subset B$ (i.e. away from a positive-codimension algebraic subscheme of $B$; this will imply $U$ has full measure) such that $f^{-1}(b)$ has property $\mathcal P$. [aside: it is known that the general fibre is a disjoint union of smooth varieties.] [second aside: this is different from the "generic fibre" in algebraic geometry, which is a scheme that is no longer over $\mathbb C$, but over a different field: some transcendental extension of $\mathbb C$.]

I. Prelude: path-connectedness

In the study of classical geometry/topology, one cares about path-connectedness. Given a topological space (e.g. manifold) $X$, two points $x,y\in X$ are path-connected if there is a continuous map $\gamma:[0,1]\to X$ with endpoints $\gamma(0)=x,\gamma(1)=y$. Noting that paths may be glued and reversed, this defines an equivalence relation on points of the manifold, and the number of path components $\#\pi_0(X)\in \mathbb N$ is a topological invariant of $X$.

Here's an informal example. Let $X$ be the collection of all smooth embeddings $S^1\to\mathbb R^3$, i.e. ways to embed a loop of string in $3$-space. Endowed with an appropriate topology, paths in $X$ should correspond to continuous movements of the string, without crossing or pinching. The path components of $X$ are then precisely the knots up to isomorphism!

Now, we pivot to algebraic geometry. The key difference: for manifolds, there is a single local building block $\mathbb R^n$; meanwhile, for varieties, there are as many different local building blocks as there are finitely generated $\mathbb C$-algebras. For example, the circle $S^1$ is the unique compact $1$-dimensional manifold; on the other hand, there are uncountably many compact Riemann surfaces. (Their topological type is indexed by the genus $g\in \mathbb Z_{\geq 0}$; but for each genus $g\geq 1$, the non-isomorphic complex structures that a genus-$g$ oriented surface may be endowed with, are indexed by points on a $\dim_{\mathbb C} = \max(1,3g-3)$ variety.)

Amidst the curves is $\mathbb P^1$, the projective line (also known to differential geometers as the Riemann sphere; also denoted $\mathbb C\mathbb P^1$ or $\mathbb C_\infty$). It is the unique genus-$0$ smooth proper curve (to a differential geometer, proper means compact, smooth curve means Riemann surface, and genus-$0$ means there are no nontrivial global holomorphic $1$-forms). In many ways, it is the algebraic analogue of the unit interval $[0,1]$. For instance, being topologically a sphere, it is also the unique simply-connected smooth proper curve. 

II. Rational-connectedness

Replacing $[0,1]$ with $\mathbb P^1$, we obtain the following definition. For a variety $X$, a rational curve on $X$ is a morphism $\mathbb P^1\to X$. $X$ is rationally connected (abbreviated rat-conn) if, for any general pair of points $x,y\in X$, there is a sequence of points $x=x_0,x_1,x_2,\dots,x_l=y$ and for each $i$, a rational curve whose image contains both $x_{i-1}$ and $x_i$. 

Now, whenever one pulls a definition from geometry into algebraic geometry, there is always the question of whether it was the "correct" definition. Here are some soft arguments for why this is the case:

• Here is why we allow for chains of images of rational curves. In topology, we have smooth deformations and continuous deformations. The analogous concepts in algebraic geometry are smooth families and flat families. It turns out that one may degenerate a smoothly embedded $\mathbb P^1$ into two copies of $\mathbb P^1$ joined by a node: a model for this is the flat family $\{xy=t\}\subset \mathbb P^2$ over $\mathbb A^1_t$. This is why we allow chains. 
• Nevertheless, the following is true: $X$ is rat-conn if and only if for any finite set of points, there is an immersed rational curve through the points, with ample normal bundle. [aside: we cannot ask for an embedding, since the only embedded rational curves in $\mathbb P^2$ are lines and conics, which are determined by $2$ and $5$ general points respectively.]
• In fact, $X$ is rat-conn if and only if there is some smooth embedded rational curve with ample normal bundle!

III. The main theorem

The key result of the paper is the following: let $X$ be a variety, and $X\to \mathbb P^1$ a non-constant morphism with rat-conn general fibre. Then the morphism admits a section $\mathbb P^1\to X$.

Here are several consequences of the main theorem:

• If a morphism $X\to B$ of varieties is dominant (i.e. image is dense), $B$ is rat-conn, and the general fibre is rat-conn, then so is $X$.
• Proving the stronger conditions above are equivalent to rational connectedness, also uses the main theorem. 

IV. Proof sketch of main theorem

As in many proofs in algebraic geometry, the core ideas are geometric in nature, but the true difficulty lies in the technical details. Here is a greatly oversimplified presentation of the proof:

1. A non-constant map $C\to \mathbb P^1$ from a smooth projective curve of genus $g\geq 1$, is a degree-$d$ covering map away from finitely many ramification points, where $d>1$. The general fibre is $d$ points, which is not rat-conn. This settles the case $\dim X=1$; we may now assume $\dim X\geq 2$. [context: this is how to build or describe such a map: take $d$ separate copies of $\mathbb P^1$, called "sheets". Mark certain points on these sheets, called "ramification points". These must be the same points on each copy of $\mathbb P^1$. Using a pair of scissors, cut along lines between the ramification points, and use these newly created edges to glue different sheets together seamlessly (the same exact cut between the same exact pair of points must be made on different sheets for them to possibly glue).]
2. Take any embedding $X\subset \mathbb P^N$, and iteratedly slice with hyperplanes to obtain an embedded curve $C\subset X$. For a general choice of slicing, we have the following: 
• $C$ is connected, smooth, and the induced map $C\to \mathbb P^1$ is a covering map away from finitely many ramification points. 
• The branch points whose fibres (of $X\to\mathbb P^1$) are smooth, have simple branching. There are only finitely many singular fibres of $X\to\mathbb P^1$; any branch point whose fibre is singular, may have arbitrarily complicated monodromy. [context: in the cut-and-paste model of a covering map with ramifications, loops around branch points at the base lift to paths that permute the sheets. Simple branching corresponds to a transposition, i.e. only two sheets are swapped. In general, the monodromy above a branch point could be any permutation of the $d$ sheets.] [second aside: one may initially believe that a general choice of slicing can avoid singular points of singular fibres, and thus avoid branch points whose fibres are singular entirely. However, this may not be possible if the singular fibres have non-reduced structure, i.e. components with multiplicity.]
3. Make $C$ pre-flexible: 
• Add rational bridges which are carefully chosen to, in the bullet point two below this one, "cancel the monodromy" about branching in singular fibres.
• Smoothen the curve; this is possible as adding sufficiently many rational tails at general points in general transverse directions makes it have ample normal bundle.
• Take an appropriate limit to remove any branching in singular fibres. 
• The limit curve may be singular; drop any vertical tails within the singular fibres. 
4. Make $C$ flexible: add rational tails at general points in general transverse directions, then smoothen it.
5. Maximally degenerate $C$ into copies of $\mathbb P^1$ joined nodally, each mapping to $\mathbb P^1$ with degree $0$ (i.e. contracted to a point) or degree $1$. 
6. Any $\mathbb P^1$-component mapping with degree-$1$ to the base $\mathbb P^1$ is an isomorphism, so its inverse defines a section $\mathbb P^1\to X$. $\quad\square$

V. Sketch of technical details involved

Now, let's pull back the curtain to reveal some of the technical details involved in formalising the argument above. 

• A non-constant map $X\to \mathbb P^1$ (or more generally, to a curve) is automatically dominant. Since $X$ is proper, so it is also automatically surjective. 
• Generic smoothness: such a map has smooth generic fibre. This will imply the general fibre is smooth. 
• Fixing a homology class $\beta\in H_2(X,\mathbb Z)$ (classical singular homology), there is a moduli stack $\overline{\mathcal M}_g(X,\beta)$ of (families of) stable maps $h:C\to X$ from at-worst-nodal genus-$g$ curves $C$, with "image-with-multiplicity" having homology class $h_*[C] = \beta$. The curve deformations are really happening in this setting. This is Gromov-Witten theory.
• $\overline{\mathcal M}_g(X,\beta)$ is a smooth proper Deligne-Mumford stack. Intuitively, this means the moduli of stable curves is a smooth orbifold; think: a compact complex manifold with at-worst-finite-group-quotient singularities. It has a closely-related "coarse moduli" scheme $\overline{M}_g(X,\beta)$, whose points are in bijection with stable genus-$g$ curves mapping to $X$ with homology class $\beta$.
• The map $X\to \mathbb P^1$ (with homology pushforward $\beta\mapsto d[\text{pt}]$) induces proper pushforward morphisms of both the coarse moduli stacks $\overline{\mathcal M}_g(X,\beta) \to \overline{\mathcal M}_g(\mathbb P^1,d)$, and the coarse moduli schemes $\overline M_g(X,\beta)\to \overline M_g(\mathbb P^1,d)$. [in this context, "proper" implies that given a stable curve on $X$, one may lift a deformation of its stable image in $\mathbb P^1$ to a deformation of the curve in $X$.]
• $\overline{\mathcal M}_g(\mathbb P^1,d)$ is irreducible, its general point is a covering map (with ramification) from a smooth genus-$g$ curve, and it also contains some "fully degenerate" point, i.e. a map from a configuration of $\mathbb P^1$'s joined nodally, each component mapping isomorphically.
• Deformation theory (Vistoli) relates the following:
• Sheaf cohomology $H^1$ of the normal bundle to $C$ twisted at sufficiently many general points in general transverse directions, vanishes. 
• The normal bundle of the new $C$ (after adding rational tails in general fibres in general directions) is (very) ample.
• (Many) first-order deformations of the new $C$ exist.
• (Many) analytic deformations of the new $C$ exist.
• (Many) algebraic deformations of the new $C$ exist.

VI. Relation to other properties

Rational connectedness is related to other notions, via the following implications: rational $\implies$ unirational $\implies$ rat-conn $\implies$ uniruled. 

Unlike the other conditions, rational connectedness behaves well in smooth families: it is both an open and a closed condition. The hope is that rational connectedness will help to study explicit varieties for which it is not known if they are (uni)rational.