@article{265,
abstract = {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.},
author = {Timothy Browning and Vishe, Pankaj},
journal = {Geometric Methods in Algebra and Number Theory},
number = {7},
pages = {1657 -- 1675},
publisher = { Mathematical Sciences Publishers},
title = {{Rational curves on smooth hypersurfaces of low degree}},
doi = {10.2140/ant.2017.11.1657},
volume = {11},
year = {2017},
}
@article{266,
abstract = {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.},
author = {Timothy Browning and Heath-Brown, Roger},
journal = {Journal of the European Mathematical Society},
number = {2},
pages = {357 -- 394},
publisher = {European Mathematical Society Publishing House},
title = {{Forms in many variables and differing degrees}},
doi = {10.4171/JEMS/668},
volume = {19},
year = {2017},
}
@article{267,
abstract = {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.},
author = {Timothy Browning},
journal = {Mathematika},
number = {3},
pages = {818 -- 839},
publisher = {Cambridge University Press},
title = {{Many cubic surfaces contain rational points}},
doi = {10.1112/S0025579317000195},
volume = {63},
year = {2017},
}
@article{268,
abstract = {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.},
author = {Timothy Browning and Prendiville, Sean M},
journal = {International Mathematics Research Notices},
number = {7},
pages = {2219 -- 2248},
publisher = {Oxford University Press},
title = {{A transference approach to a Roth-type theorem in the squares}},
doi = {10.1093/imrn/rnw096},
volume = {2017},
year = {2017},
}
@article{269,
author = {Browning, Timothy D and Loughran, Daniel},
journal = {Mathematische Zeitschrift},
number = {3-4},
pages = {1249 -- 1267},
publisher = {Springer},
title = {{Varieties with too many rational points}},
doi = {10.1007/s00209-016-1746-2},
volume = {285},
year = {2017},
}