Density of rational points (Brendan Hassett, Rice)

We'll discuss examples of varieties with dense rational points, and
how the rational points are produced.  The main goal is to introduce
useful techniques, rather than to give complete proofs in each case.

Despite the appearance of this outline, I plan to be pretty informal,
introducing the students to the main ideas without being terribly
rigorous or systematic.

1)Brauer-Severi varieties and quadric hypersurfaces
	-Geometric introduction
	-If there is a k-rational point, the variety is k-rational

2)Rational surfaces
	-A smooth cubic surface with a k-rational point is k-unirational
	-Resum'e of the classification of rational surfaces
        -Quintic Del Pezzo surfaces are always k-rational
        -Quartic Del Pezzo reduced to cubic surfaces
        -Open problems on degree-one Del Pezzo surfaces
References:
	Manin, Cubic Forms: Algebra, geometry, and arithmetic
	Koll'ar, Unirationality of cubic hypersurfaces.  J. Inst. Math.
		Jussieu  1  (2002),  no. 3, 467--476
	Skorobogatov, Torsors and rational points,
        	Cambridge Tracts in Mathematics, 144.
        	Cambridge University Press, Cambridge, 2001
        Skorobogatov, On a theorem of Enriques-Swinnerton-Dyer,
        	Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), no. 3, 429--440.

3)Higher-dimensional Fano varieties
	-Cubic hypersurfaces
	-Quartic hypersurfaces
	-Arbitrary degree hypersurfaces of large dimension
	-Open problems on sextic double solids
References:
        Harris and Tschinkel, Rational points on quartics,
        	Duke Math. J. 104 (2000), no. 3, 477--500.
	Harris, Mazur, and Pandharipande, Hypersurfaces of low degree.
		Duke Math. J. 95 (1998), no. 1, 125--160.

4)Surfaces with trivial canonical class
	-Abelian varieties
	-Quartic surfaces containing a line
	-Elliptic K3 surfaces
References:
	Tschinkel and Bogomolov, On the density of rational points on
		elliptic fibrations, J. Reine Angew. Math. 511 (1999),
		87--93.
	Hassett, Potential density of rational points on algebraic
		varieties.  Higher dimensional varieties and rational points
		(Budapest, 2001),  223--282, Bolyai Soc. Math. Stud., 12,
		Springer, Berlin, 2003.

5)Rationally connected varieties over function fields
	-Geometric interpretations in terms of sections of fibrations
	-Density and weak approximation
	-Opne problems: Cubic surfaces
References:
	Koll'ar, Miyaoka, Mori, Rationally connected varieties,
		J. Algebraic Geom. 1 (1992), no. 3, 429--448.
	Graber, Harris, Starr, Families of rationally connected varieties.
		J. Amer. Math. Soc. 16 (2003), no. 1, 57--67
	Hassett and Tschinkel, Weak approximation over function fields,
		preprint