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CATEGORIES:Number Theory Seminar
SUMMARY:Density of rational points on del Pezzo surfaces o
f degree 1 - Rosa Winter (King's College London)
DTSTART;TZID=Europe/London:20221122T143000
DTEND;TZID=Europe/London:20221122T153000
UID:TALK180419AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/180419
DESCRIPTION:Let X be an algebraic variety over an infinite fie
ld k. In arithmetic geometry we are interested in
the set X(k) of k-rational points on X. For\nexamp
le\, 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 classif
ied by their degree d\, which is an integer betwee
n 1 and 9 (for d ≥ 3\, these are the smooth surfac
es of degree d\nin P^d\n). For del Pezzo surfaces
of degree at least 2 over a field k\, we know\ntha
t the set of k-rational points is Zariski dense pr
ovided that the surface\nhas one k-rational point
to start with (that lies outside a specific subset
\nof 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 o
f k-rational points is Zariski\ndense in general.\
nI will talk about density of rational points on d
el Pezzo surfaces\, state what\nis known so far\,
and show a result that is joint work with Julie De
sjardins\,\nin which we give sufficient and necess
ary conditions for the set of k-rational\npoints o
n a specific family of del Pezzo surfaces of degre
e 1 to be Zariski\ndense\, where k is finitely gen
erated over Q.
LOCATION:Centre for Mathematical Sciences\, MR13
CONTACT:Rong Zhou
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