\begin{thebibliography}{GHS03}

\bibitem[Bak08]{Baker}
H.~F. Baker.
\newblock On the {I}nvariants of a {B}inary {Q}uintic and the {R}eality of its
  {R}oots.
\newblock {\em Proc. London Math. Soc.}, S2-6(1):122, 1908.

\bibitem[Bea77]{BeauvillePrym}
Arnaud Beauville.
\newblock Vari\'et\'es de {P}rym et jacobiennes interm\'ediaires.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup. (4)}, 10(3):309--391, 1977.

\bibitem[Bru78]{Brumer}
Armand Brumer.
\newblock Remarques sur les couples de formes quadratiques.
\newblock {\em C. R. Acad. Sci. Paris S\'er. A-B}, 286(16):A679--A681, 1978.

\bibitem[Cas04]{Castravet}
Ana-Maria Castravet.
\newblock Rational families of vector bundles on curves.
\newblock {\em Internat. J. Math.}, 15(1):13--45, 2004.

\bibitem[Che06]{Cheltsov}
Ivan Cheltsov.
\newblock Nonrational nodal quartic threefolds.
\newblock {\em Pacific J. Math.}, 226(1):65--81, 2006.

\bibitem[DI87]{DI}
Pierre Deligne and Luc Illusie.
\newblock Rel\`evements modulo {$p^2$} et d\'ecomposition du complexe de de
  {R}ham.
\newblock {\em Invent. Math.}, 89(2):247--270, 1987.

\bibitem[DIM11]{DIM2}
Olivier Debarre, Atanas Iliev, and Laurent Manivel.
\newblock On nodal prime {F}ano threefolds of degree 10.
\newblock {\em Sci. China Math.}, 54(8):1591--1609, 2011.

\bibitem[DIM12]{DIM}
Olivier Debarre, Atanas Iliev, and Laurent Manivel.
\newblock On the period map for prime {F}ano threefolds of degree 10.
\newblock {\em J. Algebraic Geom.}, 21(1):21--59, 2012.

\bibitem[dJS03]{dJS}
A.~J. de~Jong and J.~Starr.
\newblock Every rationally connected variety over the function field of a curve
  has a rational point.
\newblock {\em Amer. J. Math.}, 125(3):567--580, 2003.

\bibitem[Don81]{DonagiBAMS}
Ron Donagi.
\newblock The tetragonal construction.
\newblock {\em Bull. Amer. Math. Soc. (N.S.)}, 4(2):181--185, 1981.

\bibitem[Esn03]{Esn}
H{\'e}l{\`e}ne Esnault.
\newblock Varieties over a finite field with trivial {C}how group of 0-cycles
  have a rational point.
\newblock {\em Invent. Math.}, 151(1):187--191, 2003.

\bibitem[GHS03]{GHS}
Tom Graber, Joe Harris, and Jason Starr.
\newblock Families of rationally connected varieties.
\newblock {\em J. Amer. Math. Soc.}, 16(1):57--67 (electronic), 2003.

\bibitem[Has09]{HasCMI}
Brendan Hassett.
\newblock Rational surfaces over nonclosed fields.
\newblock In {\em Arithmetic geometry}, volume~8 of {\em Clay Math. Proc.},
  pages 155--209. Amer. Math. Soc., Providence, RI, 2009.

\bibitem[HH13]{HH}
Brendan Hassett and Donghoon Hyeon.
\newblock Log minimal model program for the moduli space of stable curves: the
  first flip.
\newblock {\em Ann. of Math. (2)}, 177(3):911--968, 2013.

\bibitem[Hos96]{Hosoh}
Toshio Hosoh.
\newblock Automorphism groups of quartic del {P}ezzo surfaces.
\newblock {\em J. Algebra}, 185(2):374--389, 1996.

\bibitem[HRS05]{HRS}
Joe Harris, Mike Roth, and Jason Starr.
\newblock Curves of small degree on cubic threefolds.
\newblock {\em Rocky Mountain J. Math.}, 35(3):761--817, 2005.

\bibitem[HT12a]{HTembed}
Brendan Hassett and Yuri Tschinkel.
\newblock Embedding pointed curves in {$K3$} surfaces, 2012.

\bibitem[HT12b]{HTquad}
Brendan Hassett and Yuri Tschinkel.
\newblock Spaces of sections of quadric surface fibrations over curves.
\newblock In {\em Compact moduli spaces and vector bundles}, volume 564 of {\em
  Contemp. Math.}, pages 227--249. Amer. Math. Soc., Providence, RI, 2012.

\bibitem[IM00]{IM}
A.~Iliev and D.~Markushevich.
\newblock The {A}bel-{J}acobi map for a cubic threefold and periods of {F}ano
  threefolds of degree 14.
\newblock {\em Doc. Math.}, 5:23--47 (electronic), 2000.

\bibitem[IM07]{IM2}
Atanas Iliev and Dimitri Markushevich.
\newblock Parametrization of sing {$\Theta$} for a {F}ano 3-fold of genus 7 by
  moduli of vector bundles.
\newblock {\em Asian J. Math.}, 11(3):427--458, 2007.

\bibitem[JHS]{HRS2}
M.~Roth J.~Harris and J.~Starr.
\newblock {A}bel-{J}acobi maps associated to smooth cubic threefolds.
\newblock arXiv:math/0202080.

\bibitem[Kan89]{Kanev}
Vassil Kanev.
\newblock Intermediate {J}acobians and {C}how groups of three-folds with a
  pencil of del {P}ezzo surfaces.
\newblock {\em Ann. Mat. Pura Appl. (4)}, 154:13--48, 1989.

\bibitem[Koi88]{Koita}
Masanori Koitabashi.
\newblock Automorphism groups of generic rational surfaces.
\newblock {\em J. Algebra}, 116(1):130--142, 1988.

\bibitem[Kol96]{KollarRC}
J{\'a}nos Koll{\'a}r.
\newblock {\em Rational curves on algebraic varieties}, volume~32.
\newblock Springer-Verlag, Berlin, 1996.

\bibitem[Lan55]{Lang}
Serge Lang.
\newblock Abelian varieties over finite fields.
\newblock {\em Proc. Nat. Acad. Sci. U.S.A.}, 41:174--176, 1955.

\bibitem[Lee]{Leep}
David~B. Leep.
\newblock The {A}mer-{B}rumer theorem over arbitrary fields.
\newblock Preprint.

\bibitem[MM93]{MM}
Toshiki Mabuchi and Shigeru Mukai.
\newblock Stability and {E}instein-{K}\"ahler metric of a quartic del {P}ezzo
  surface.
\newblock In {\em Einstein metrics and {Y}ang-{M}ills connections ({S}anda,
  1990)}, volume 145 of {\em Lecture Notes in Pure and Appl. Math.}, pages
  133--160. Dekker, New York, 1993.

\bibitem[MT01]{MT}
D.~Markushevich and A.~S. Tikhomirov.
\newblock The {A}bel-{J}acobi map of a moduli component of vector bundles on
  the cubic threefold.
\newblock {\em J. Algebraic Geom.}, 10(1):37--62, 2001.

\bibitem[Muk95]{MukAJM}
Shigeru Mukai.
\newblock Curves and symmetric spaces. {I}.
\newblock {\em Amer. J. Math.}, 117(6):1627--1644, 1995.

\bibitem[Nam97]{Namikawa}
Yoshinori Namikawa.
\newblock Smoothing {F}ano {$3$}-folds.
\newblock {\em J. Algebraic Geom.}, 6(2):307--324, 1997.

\bibitem[Rei72]{Reid}
Miles Reid.
\newblock {\em The complete intersection of two or more quadrics}.
\newblock PhD thesis, Cambridge University, 1972.

\bibitem[Sha78]{Shatz}
Stephen~S. Shatz.
\newblock On subbundles of vector bundles over {${\bf P}^{1}$}.
\newblock {\em J. Pure Appl. Algebra}, 10(3):315--322, 1977/78.

\bibitem[Shr06]{Shramov}
K.~A. Shramov.
\newblock On the rationality of nonsingular threefolds with a pencil of del
  {P}ezzo surfaces of degree 4.
\newblock {\em Mat. Sb.}, 197(1):133--144, 2006.

\bibitem[Tju75]{Tjurin}
A.~N. Tjurin.
\newblock The intersection of quadrics.
\newblock {\em Uspehi Mat. Nauk}, 30(6(186)):51--99, 1975.

\bibitem[Zhu11]{Zhu}
Yi~Zhu.
\newblock Homogeneous fibrations over curves, 2011.
\newblock arXiv:1111.2963v2.

\end{thebibliography}