Maximal rationally connected fibrations and movable curves

Smooth rationally connected threefolds contain all smooth curves

It is very easy to see that every smooth projective curve can be embedded in P3. Eisenbud and Harris asked whether the same is true if P3 is replaced by an arbitrary smooth rational projective 3-fold X and Eisenbud suggested starting with the case where X is toric. In that case the answer is yes, and one can see that in a very explicit way, as was done in my preprint [Sa2]. In response to [Sa2]...

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...

A note on superspecial and maximal curves

In this note we review a simple criterion, due to Ekedahl, for superspecial curves defined over finite fields.Using this we generalize and give some simple proofs for some well-known superspecial curves.