Preprints

Big semistable vector bundles

Preprint, 2002. This preprint formulates a conjecture on big semistable vector bundles on projective varieties. It also proves the conjecture for vector bundles over curves. Work supported by NSF grant DMS-0200892.

A quantitative proof of Roth's theorem with moving targets

Preprint, 1995. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. This paper was formerly titled Roth's theorem with moving targets.

On McQuillan's "tautological inequality" and the Weyl-Ahlfors theory of associated curves

Preprint, 2007; arXiv:0706.3044.

Nagata's embedding theorem

Preprint, 2007; arXiv:0706.1907.

Transplanting Faltings' garden

Preprint, 2009; arXiv:0901.2106.

Publications

A higher dimensional Mordell conjecture

In: Arithmetic Geometry, ed. by G. Cornell and J. H. Silverman, Springer-Verlag, New York, 1986, pp. 341-353. MR 89b:14029 (whole collection); Zbl. 605.14019.

A diophantine conjecture over Q

In: Seminaire de Theorie des Nombres, Paris 1984-85, ed. by Catherine Goldstein, Progress in Mathematics 63, Birkhauser, Boston-Basel-Stuttgart, 1986, pp. 241-250. MR 88h:11045; Zbl. 601.14016.

Examples of some Q-admissible groups (joint with W. Feit)

J. of Number Theory 26 (1987), pp. 210-226. Unfortunately, this paper contains a serious error which invalidates the main result. MR 88g:12006; Zbl. 619.12007.

Diophantine approximations and value distribution theory

Lecture Notes in Mathematics 1239, Springer-Verlag, New York, 1987. 132+x pp. MR 91k:11049; Zbl. 609.14011.
List of errata and addenda: dvi, pdf.
A scanned copy of the book is now available from springerlink (access restricted, except for front matter and back matter).

A refinement of Schmidt's subspace theorem

The American Journal of Mathematics, 111 (1989), pp. 489-518. MR 90f:11054; Zbl. 662.14002.

Dyson's lemma for products of two curves of arbitrary genus

Invent. Math. 98 (1989), pp. 107-113. MR 90k:11075; Zbl. 666.10024.

Mordell's conjecture over function fields

Invent. Math. 98 (1989), pp. 115-138. MR 90k:11076; Zbl. 662.14019.

Diophantine inequalities and Arakelov theory

In: S. Lang, Introduction to Arakelov Theory, Springer, 1988, pp. 155-178. MR 89m:11059 (whole book); Zbl. 667.14001 (whole book).

Arithmetic discriminants and quadratic points on curves

In: G. van der Geer, F. Oort, and J. H. M. Steenbrink, eds., Arithmetic algebraic geometry, Texel 1989, Birkhauser, Boston, 1991, pp. 359-376. MR 92j:11059; Zbl. 749.14018.

On algebraic points on curves

Comp. Math. 78 (1991), pp. 29-36. MR 93b:11080; Zbl. 731.14015.

Siegel's theorem in the compact case

Ann. Math. 133 (1991), pp. 509-548. MR 93d:11065; Zbl. 774.14019.

Recent work on Nevanlinna theory and Diophantine approximation

In: W. Stoll, ed., Proceedings Symposium on Value Theory in Several Complex Variables, Notre Dame, Indiana, April, 1990, University of Notre Dame Press, Notre Dame, 1992, pp. 107-113. MR 95c:11095; Zbl. 871.11043.

Arithmetic and hyperbolic geometry

In: Proceedings of the International Congress of Mathematicians, Kyoto, Japan, August 1990, Springer, Tokyo, 1991, pp. 757-765. MR 93e:11080; Zbl. 745.14007.

A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing

Journal of the AMS 5 (1992), pp. 763-804. MR 94a:11093; Zbl. 778.11037.

Arithmetic of Subvarieties of Abelian and Semiabelian Varieties

In: Advances in Number Theory (Proceedings of the Canadian Number Theory Association, Queens University, Kingston, Ontario, August, 1991), Fernando Q. Gouvea and Noriko Yui, eds., Clarendon Press, Oxford, 1993, pp. 233-238. MR 97a:11101; Zbl. 790.11048.

Applications of arithmetic algebraic geometry to diophantine approximations

In: Arithmetic Algebraic Geometry, Trento, 1991, Lecture Notes in Mathematics 1553, Springer-Verlag, Heidelberg, 1993, pp. 164-208. This is available as a TeX file, which in turn requires a special macro file PVmacs.sln. MR 96c:11067; Zbl. 846.14009 (individual article) and 780.00022 (whole book).

Roth's theorem with moving targets

International Mathematics Research Notices 1996 (1996), pp. 109-114. An earlier version of this paper had the title Roth's theorem with moving targets - the sequel. MR 96k:11087; Zbl. 877.11041.

Integral points on subvarieties of semiabelian varieties, I

Inventiones Mathematicae 126 (1996), pp. 133-181. MR 98a:14034; Zbl. 1011.11040.

Schmidt's Subspace Theorem with moving targets (joint with Min Ru)

Inventiones Mathematicae 127 (1997), pp. 51-65. MR 97g:11076; Zbl. 1013.11044.

On Cartan's Theorem and Cartan's Conjecture

The American Journal of Mathematics 119 (1997), pp. 1-17. MR 97m:32041; Zbl. 877.11040.

A more general abc conjecture

International Mathematics Research Notices 1998 (1998), pp. 1103-1116. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 99k:11096; Zbl. 923.11059.

Integral points on subvarieties of semiabelian varieties, II

The American Journal of Mathematics 121 (1999), pp. 283-313. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 2000d:11074; Zbl. 1018.11027.

Nevanlinna theory and diophantine approximation

In: Several complex variables (Math. Sci. Res. Inst. Publ. #37), Michael Schneider and Yum-Tong Siu, eds., Cambridge U. Press, New York, 1999, pp. 535-564. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 2001j:11072; Zbl. 960.32013.

On the abc conjecture and diophantine approximation by rational points

The American Journal of Mathematics 122 (2000), pp. 843-872. This is available as a TeX file, which also requires a MetaPost file, in addition to the generic macro file PVmacs.tex. MR 2001i:11094; Zbl. 1037.11052.

Diagonal quadratic forms and Hilbert's tenth problem

In: Hilbert's tenth problem: relations with arithmetic and algebraic geometry (Ghent, Belgium, 1999) (Contemporary Mathematics #270), Jan Denef, Leonard Lipshitz, Thanases Pheidas, Jan Van Geel, eds., American Mathematical Society, Providence, R.I., 2000, pp. 261-274. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 2001k:11260; Zbl. 0995.11070.

Correction to "On the abc conjecture and diophantine approximation by rational points"

The American Journal of Mathematics 123 (2001), pp. 383-384. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 2002d:11095; Zbl. 1037.11053.

Arithmetic jet spaces

In: Proceedings of Hayama Symposium on Several Complex Variables 2004, December 18--21, ed. by Y. Nishimura, et al., Shonan Village, Hayama, Japan, 2005, pp. 134--143.

On the Nochka-Chen-Ru-Wong proof of Cartan's Conjecture

Journal of Number Theory 125 (2007), pp. 229--234. Gives a mild shortening of the construction of weights associated to hyperplanes in Nochka's proof of the Cartan conjecture on holomorphic curves approximating hyperplanes in n-subgeneral position. Work supported by NSF grants DMS-9304899, DMS-0200892, and DMS-0500512.

Jets via Hasse-Schmidt derivations

In: Diophantine Geometry, Proceedings, U. Zannier (ed.), Edizioni della Normale, Pisa, 2007, pp. 335-361. This note gives an expository introduction to the theory of jet spaces on arbitrary schemes, defined using Hasse-Schmidt derivations. It was written as a part of the seminar on motivic integration taking place at MSRI. Work supported by NSF grant DMS-0200892.

Diophantine approximation and Nevanlinna theory

In: Arithmetic Geometry, Cetraro, Italy 2007, Lecture Notes in Mathematics 2009, Springer-Verlag, Berlin Heidelberg, 2011, pp. 111-230.

Multiplier ideal sheaves, Nevanlinna theory, and diophantine approximation

In: Number theory, analysis and geometry: In memory of Serge Lang, ed. by Dorian Goldfeld, Jay Jorgenson, Peter Jones, Dinakar Ramakrishnan, Kenneth A. Ribet, and John Tate, Springer, New York, 2012, pp. 647-658. arXiv:0709.3322.