TY - JOUR
AB - We establish the dimension and irreducibility of the moduli space of rational curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low degree. A spreading out argument reduces the problem to hypersurfaces defined over finite fields of large cardinality, which can then be tackled using a function field version of the Hardy-Littlewood circle method, in which particular care is taken to ensure uniformity in the size of the underlying finite field.
AU - Timothy Browning
AU - Vishe, Pankaj
ID - 265
IS - 7
JF - Geometric Methods in Algebra and Number Theory
TI - Rational curves on smooth hypersurfaces of low degree
VL - 11
ER -
TY - JOUR
AB - We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a smooth and geometrically integral variety X Pm, provided only that its dimension is large enough in terms of its degree.
AU - Timothy Browning
AU - Heath-Brown, Roger
ID - 266
IS - 2
JF - Journal of the European Mathematical Society
TI - Forms in many variables and differing degrees
VL - 19
ER -
TY - JOUR
AB - Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.
AU - Timothy Browning
ID - 267
IS - 3
JF - Mathematika
TI - Many cubic surfaces contain rational points
VL - 63
ER -
TY - JOUR
AB - We show that any subset of the squares of positive relative upper density contains nontrivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting.
AU - Timothy Browning
AU - Prendiville, Sean M
ID - 268
IS - 7
JF - International Mathematics Research Notices
TI - A transference approach to a Roth-type theorem in the squares
VL - 2017
ER -
TY - JOUR
AU - Browning, Timothy D
AU - Loughran, Daniel
ID - 269
IS - 3-4
JF - Mathematische Zeitschrift
TI - Varieties with too many rational points
VL - 285
ER -