Rationally connected varieties over local fields.

*(English)*Zbl 0976.14016Let \(X\) be a proper variety defined over a field \(K\). Following Yu. I. Manin [Cubic forms. Algebra, geometry, arithmetic. Transl. from the Russian by M. Hazewinkel. 2nd ed. North-Holland Mathematical Library, Vol. 4 (1986; Zbl 0582.14010), we say that two points \(x,x'\in X(K)\) are \(R\)-equivalent if they can be connected by a chain of rational curves defined over \(K\). \(R\)-equivalence is only interesting if there are plenty of rational curves on \(X\), at least over \(\overline K\). There are many a priori different ways of defining what “plenty” of rational curves should mean. Fortunately many of these turn out to be equivalent and this leads to the notion of rationally connected varieties:

Definition – theorem 1.1. Let \(\overline K\) be an algebraically closed field of characteristic zero. A smooth proper variety \(X\) over \(K\) is called rationally connected if it satisfies any of the following equivalent properties:

1. There is an open subset \(\emptyset\neq U\subset X\), such that for every \(x_1,x_2\in U\), there is a morphism \(f:\mathbb{P}^1\to X\) satisfying \(x_1,x_2\in f(\mathbb{P}^1)\).

2. For every \(x_1,x_2\in X\), there is a morphism \(f:\mathbb{P}^1\to X\) satisfying \(x_1,x_2\in f(\mathbb{P}^1)\).

3. For every \(x_1,\dots, x_n\in X\), there is a morphism \(f:\mathbb{P}^1\to X\) satisfying \(x_1, \dots, x_n\in f(\mathbb{P}^1)\).

4. Let \(p_1,\dots,p_n\in\mathbb{P}^1\) be distinct points and \(m_1,\dots,m_n\) natural numbers. For each \(i\) let \(f_i:\text{Spec} \overline K[t]/(t^{m_i})\to X\) be a morphism. Then there is a morphism \(f: \mathbb{P}^1 \to X\) such that the Taylor series of \(f\) at \(p_i\) coincides with \(f_i\) up to order \(m_i\) for every \(i\).

5. There is a morphism \(f:\mathbb{P}^1\to X\) such that \(f^*T_X\) is ample.

6. For every \(x_1,\dots, x_n\in X\) there is a morphism \(f:\mathbb{P}^1\to X\) such that \(f^*T_X\) is ample and \(x_1,\dots, x_n\in f(\mathbb{P}^1)\).

The situation is somewhat more complicated in positive characteristic. The conditions of (1.1) are not mutually equivalent, but it turns out that (1.1.5) implies the rest [see J. Kollár, “Rational curves on algebraic varieties”. (1995; Zbl 0877.14012)]). Such varieties are called separably rationally connected. The weakness of this notion is that there are even unirational varieties which are not separably rationally connected, and thus we do not cover all cases where finiteness is expected. – In characteristic zero, the class of rationally connected varieties is closed under smooth deformations and it contains all the known “rational like” varieties. For instance, unirational varieties and Fano varieties are rationally connected.

By a local field \(K\), we mean either \(\mathbb{R},\mathbb{C},\mathbb{F}_q((t))\) or a finite extension of the \(p\)-adic field \(\mathbb{Q}_p\). Each of these fields has a natural locally compact topology, and this induces a locally compact topology on the \(K\)-points of any algebraic variety over \(K\), called the \(K\)-topology. The \(K\)-points of a proper variety are compact in the \(K\)-topology. – The aim of this paper is to study the \(R\)-equivalence classes on rationally connected varieties over local fields. The main result shows the existence of many rational curves defined over \(K\). This in turn implies that there are only finitely many \(R\)-equivalence classes.

Theorem 1.4. Let \(K\) be a local field and \(X\) a smooth proper variety over \(K\) such that \(X_{\overline K}\) is separably rationally connected. Then, for every \(x\in X(K)\), there is a morphism \(f_x: \mathbb{P}^1\to X\) (defined over \(K)\) such that \(f^*_xT_X\) is ample and \(x\in F_x(\mathbb{P}^1 (K))\).

Corollary 1.5. Let \(K\) be a local field and \(X\) a smooth proper variety over \(K\) such that \(X_{\overline K}\) is separably rationally connected. Then:

1. Every \(R\)-equivalence class in \(X(K)\) is open and closed in the \(K\)-topology.

2. There are only finitely many \(R\)-equivalence classes in \(X(K)\).

It is interesting to note that such a result should characterize rationally connected varieties.

Conjecture 1.6. Let \(X\) be a smooth proper variety defined over a local field \(K\) of characteristic zero. Assume that \(X(K)\neq\emptyset\) and there are only finitely many \(R\)-equivalence classes on \(X(K)\). Then \(X\) is rationally connected.

This conjecture was proved recently. In the real case we can establish a precise relationship between the Euclidean topology of \(X\) and the \(R\)-equivalence classes:

Corollary 1.7. Let \(X\) be a smooth proper variety over \(\mathbb{R}\) such that \(X_\mathbb{C}\) is rationally connected. Then the \(R\)-equivalence classes are precisely the connected components of \(X(\mathbb{R})\).

Corollary 1.8. Let \(K\) be a local field of characteristic zero and \(X\) a smooth proper variety over \(K\). Assume that there is a morphism \(f:X\to\mathbb{P}^1\) whose geometric generic fiber \(F\) is either:

1. a Del Pezzo surface of degree \(\geq 2\),

2. a cubic hypersurface,

3. a complete intersection of two quadrics in \(\mathbb{P}^n\) for \(n\geq 4\), or

4. there is a connected linear algebraic group acting on \(F\) with a dense orbit.

Then \(X\) is unirational over \(K\) if and only if \(X(K)\neq\emptyset\).

We also obtain a weaker result over global fields:

Corollary 1.9. Let \({\mathcal O}\) be the ring of integers in a number field and \(X\) a smooth proper variety defined over \({\mathcal O}\) satisfying one of the conditions (1.8.1-1.8.4). Assume that \(X({\mathcal O})\neq \emptyset\). Then the mod \(P\) reduction of \(X\) is unirational over \({\mathcal O}/P\) for almost all prime ideals \(P< {\mathcal O}\).

Definition – theorem 1.1. Let \(\overline K\) be an algebraically closed field of characteristic zero. A smooth proper variety \(X\) over \(K\) is called rationally connected if it satisfies any of the following equivalent properties:

1. There is an open subset \(\emptyset\neq U\subset X\), such that for every \(x_1,x_2\in U\), there is a morphism \(f:\mathbb{P}^1\to X\) satisfying \(x_1,x_2\in f(\mathbb{P}^1)\).

2. For every \(x_1,x_2\in X\), there is a morphism \(f:\mathbb{P}^1\to X\) satisfying \(x_1,x_2\in f(\mathbb{P}^1)\).

3. For every \(x_1,\dots, x_n\in X\), there is a morphism \(f:\mathbb{P}^1\to X\) satisfying \(x_1, \dots, x_n\in f(\mathbb{P}^1)\).

4. Let \(p_1,\dots,p_n\in\mathbb{P}^1\) be distinct points and \(m_1,\dots,m_n\) natural numbers. For each \(i\) let \(f_i:\text{Spec} \overline K[t]/(t^{m_i})\to X\) be a morphism. Then there is a morphism \(f: \mathbb{P}^1 \to X\) such that the Taylor series of \(f\) at \(p_i\) coincides with \(f_i\) up to order \(m_i\) for every \(i\).

5. There is a morphism \(f:\mathbb{P}^1\to X\) such that \(f^*T_X\) is ample.

6. For every \(x_1,\dots, x_n\in X\) there is a morphism \(f:\mathbb{P}^1\to X\) such that \(f^*T_X\) is ample and \(x_1,\dots, x_n\in f(\mathbb{P}^1)\).

The situation is somewhat more complicated in positive characteristic. The conditions of (1.1) are not mutually equivalent, but it turns out that (1.1.5) implies the rest [see J. Kollár, “Rational curves on algebraic varieties”. (1995; Zbl 0877.14012)]). Such varieties are called separably rationally connected. The weakness of this notion is that there are even unirational varieties which are not separably rationally connected, and thus we do not cover all cases where finiteness is expected. – In characteristic zero, the class of rationally connected varieties is closed under smooth deformations and it contains all the known “rational like” varieties. For instance, unirational varieties and Fano varieties are rationally connected.

By a local field \(K\), we mean either \(\mathbb{R},\mathbb{C},\mathbb{F}_q((t))\) or a finite extension of the \(p\)-adic field \(\mathbb{Q}_p\). Each of these fields has a natural locally compact topology, and this induces a locally compact topology on the \(K\)-points of any algebraic variety over \(K\), called the \(K\)-topology. The \(K\)-points of a proper variety are compact in the \(K\)-topology. – The aim of this paper is to study the \(R\)-equivalence classes on rationally connected varieties over local fields. The main result shows the existence of many rational curves defined over \(K\). This in turn implies that there are only finitely many \(R\)-equivalence classes.

Theorem 1.4. Let \(K\) be a local field and \(X\) a smooth proper variety over \(K\) such that \(X_{\overline K}\) is separably rationally connected. Then, for every \(x\in X(K)\), there is a morphism \(f_x: \mathbb{P}^1\to X\) (defined over \(K)\) such that \(f^*_xT_X\) is ample and \(x\in F_x(\mathbb{P}^1 (K))\).

Corollary 1.5. Let \(K\) be a local field and \(X\) a smooth proper variety over \(K\) such that \(X_{\overline K}\) is separably rationally connected. Then:

1. Every \(R\)-equivalence class in \(X(K)\) is open and closed in the \(K\)-topology.

2. There are only finitely many \(R\)-equivalence classes in \(X(K)\).

It is interesting to note that such a result should characterize rationally connected varieties.

Conjecture 1.6. Let \(X\) be a smooth proper variety defined over a local field \(K\) of characteristic zero. Assume that \(X(K)\neq\emptyset\) and there are only finitely many \(R\)-equivalence classes on \(X(K)\). Then \(X\) is rationally connected.

This conjecture was proved recently. In the real case we can establish a precise relationship between the Euclidean topology of \(X\) and the \(R\)-equivalence classes:

Corollary 1.7. Let \(X\) be a smooth proper variety over \(\mathbb{R}\) such that \(X_\mathbb{C}\) is rationally connected. Then the \(R\)-equivalence classes are precisely the connected components of \(X(\mathbb{R})\).

Corollary 1.8. Let \(K\) be a local field of characteristic zero and \(X\) a smooth proper variety over \(K\). Assume that there is a morphism \(f:X\to\mathbb{P}^1\) whose geometric generic fiber \(F\) is either:

1. a Del Pezzo surface of degree \(\geq 2\),

2. a cubic hypersurface,

3. a complete intersection of two quadrics in \(\mathbb{P}^n\) for \(n\geq 4\), or

4. there is a connected linear algebraic group acting on \(F\) with a dense orbit.

Then \(X\) is unirational over \(K\) if and only if \(X(K)\neq\emptyset\).

We also obtain a weaker result over global fields:

Corollary 1.9. Let \({\mathcal O}\) be the ring of integers in a number field and \(X\) a smooth proper variety defined over \({\mathcal O}\) satisfying one of the conditions (1.8.1-1.8.4). Assume that \(X({\mathcal O})\neq \emptyset\). Then the mod \(P\) reduction of \(X\) is unirational over \({\mathcal O}/P\) for almost all prime ideals \(P< {\mathcal O}\).

##### MSC:

14G20 | Local ground fields in algebraic geometry |

14M20 | Rational and unirational varieties |

14L30 | Group actions on varieties or schemes (quotients) |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |