RATIONAL POINTS ON SMOOTH SURFACES IN <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://

Abstract We improve a bound due to Voloch [‘Surfaces in <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> $\mathbb {P}^3$ double struck upper P cubed over finite fields’, in: Topics in Algebraic and Noncommutative Geometry , Contemporary Mathematics, 324 (eds. C. G. Melles, J.-P. Brasselet, G. Kennedy, K. Lauter and L. McEwan) (American Mathematical Society, Providence, RI, 2003), 219–226] on the number of rational points on smooth surfaces in <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> $\mathbb {P}^3$ double struck upper P cubed over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact number of rational points. These computations may have independent interest.