Arithmetic toric varieties. Issue 2 (4th September 2013)

• Record Type: Journal Article
• Authors: Elizondo, E. Javier; Lima‐Filho, Paulo; Sottile, Frank; Teitler, Zach
• Is Part Of: Mathematische Nachrichten. Volume 287:Issue 2/3(2014)
• Page Start: 216
• Page End: 241
• Publication Date: 2013-09-04
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201200305
• Languages: English
• ISSNs: 0025-584X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>We study toric varieties over a field <italic>k</italic> that split in a Galois extension <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4sq49qt1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0025584X:mana201200305:equation:mana201200305-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>K</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></alternatives> using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify <italic>k</italic>‐forms of projective spaces when <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4sq49qvk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0025584X:mana201200305:equation:mana201200305-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>K</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></alternatives> is cyclic, and we also study <italic>k</italic>‐forms of surfaces.</p> </abstract>