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SUMMARY:Density of rational points on del Pezzo surfaces of degree 1 - Ros
 a Winter (King's College London)
DTSTART:20221122T143000Z
DTEND:20221122T153000Z
UID:TALK180419@talks.cam.ac.uk
CONTACT:Rong Zhou
DESCRIPTION:Let X be an algebraic variety over an infinite field k. In ari
 thmetic geometry we are interested in the set X(k) of k-rational points on
  X. For\nexample\, is X(k) empty or not? And if it is not empty\, is X(k) 
 dense in X\nwith respect to the Zariski topology?\nDel Pezzo surfaces are 
 surfaces classified by their degree d\, which is an integer between 1 and 
 9 (for d ≥ 3\, these are the smooth surfaces of degree d\nin P^d\n). For
  del Pezzo surfaces of degree at least 2 over a field k\, we know\nthat th
 e set of k-rational points is Zariski dense provided that the surface\nhas
  one k-rational point to start with (that lies outside a specific subset\n
 of the surface for degree 2). However\, for del Pezzo surfaces of degree 1
 \nover a field k\, even though we know that they always contain at least o
 ne\nk-rational point\, we do not know if the set of k-rational points is Z
 ariski\ndense in general.\nI will talk about density of rational points on
  del Pezzo surfaces\, state what\nis known so far\, and show a result that
  is joint work with Julie Desjardins\,\nin which we give sufficient and ne
 cessary conditions for the set of k-rational\npoints on a specific family 
 of del Pezzo surfaces of degree 1 to be Zariski\ndense\, where k is finite
 ly generated over Q.
LOCATION:Centre for Mathematical Sciences\, MR13
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