Varieties connected by chains of lines

Rationally connected varieties

The aim of these notes is to provide an introduction to the theory of rationally connected varieties, as well as to discuss a recent result by T. Graber, J. Harris and J. Starr.

Convex Rationally Connected Varieties

Rationally Connected Varieties

1 13 4. Free rational curves 14 5. Uniruledness 16 5.1. Definitions equivalent to uniruledness 16 5.2. Examples and consequences of uniruledness 18 6. Rational connectivity 20 6.1. Definitions equivalent to rational connectivity 21 6.2. Rational chain-connectivity implies rational connectivity 22 6.3. Examples and consequences of rational connectivity 26 7. The MRC quotient 27 7.1. Quotients by...

Families of Rationally Connected Varieties

1.1. Statement of results. We will work throughout over the complex numbers, so that the results here apply over any algebraically closed field of characteristic 0. Recall that a proper variety X is said to be rationally connected if two general points p, q ∈ X are contained in the image of a map g : P → X . This is clearly a birationally invariant property. When X is smooth, this turns out to ...

On Varieties Which Are Uniruled by Lines

Using the ♯-minimal model program of uniruled varieties we show that for any pair (X,H) consisting of a reduced and irreducible variety X of dimension k ≥ 3 and a globally generated big line bundle H on X with d := H and n := h(X,H) − 1 such that d < 2(n− k)− 4, then X is uniruled of H-degree one, except if (k, d, n) = (3, 27, 19) and a ♯-minimal model of (X,H) is (P3,OP3(3)). We also show that...