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SUMMARY:Arda Huseyin Demirhan (University of Illinois at Chicago)
DTSTART:20210319T150000Z
DTEND:20210319T160000Z
DTSTAMP:20260423T023055Z
UID:zorp_1729/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/zorp_1729/7/
 ">Distribution of Rational Points on Toric Varieties – A Multi-Height Ap
 proach</a>\nby Arda Huseyin Demirhan (University of Illinois at Chicago) a
 s part of ZORP (zoom on rational points)\n\n\nAbstract\nManin's conjecture
  was verified by Victor Batyrev and Yuri Tschinkel for toric varieties. Em
 manuel Peyre has proposed two notions\, "freeness" and "all the heights" a
 pproach to delete accumulating subvarieties in "Libert\\'e et accumulation
 " and "Beyond heights: slopes and distribution of rational points". Based 
 on the all the heights approach\, in this talk\, we will explain a multi-h
 eight variant of the Batyrev-Tschinkel theorem where one considers working
  at {\\em height boxes}\, instead of a single height function\, as a way t
 o get rid of accumulating subvarieties. This is our main result: Let $X$ b
 e an arbitrary  toric variety over a number field $F$\, and let $H_i$\, $1
  \\leq i \\leq r$\, be height functions associated to the generators of th
 e cone of effective divisors of $X$. Fix positive real numbers $a_i$\, $1 
 \\leq i \\leq r$. Then the number of rational points $P \\in X(F)$ such th
 at for each $i$\, $H_i(P) \\leq B^{a_i}$  as $B$ gets large is equal to $C
  B^{a_1 + \\dots + a_r} + O(B^{a_1 + \\dots + a_r-\\epsilon})$ for an $\\e
 psilon >0$. Our result is a first example of a large family of varieties a
 long the lines of Peyre's idea.\n
LOCATION:https://researchseminars.org/talk/zorp_1729/7/
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