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SUMMARY:Jackson Morrow (UC Berkeley)
DTSTART:20220914T210000Z
DTEND:20220914T223000Z
DTSTAMP:20260423T023044Z
UID:UBC_NTS/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UBC_NTS/1/">
 Boundedness of hyperbolic varieties</a>\nby Jackson Morrow (UC Berkeley) a
 s part of UBC (online) Number Theory Seminar\n\n\nAbstract\nLet $C_1$\, $C
 _2$ be smooth projective curves over an algebraically closed field $K$ of 
 characteristic zero. What is the behavior of the set of non-constant maps 
 $C_1 \\to C_2$? Is it infinite\, finite\, or empty? It turns out that the 
 answer to this question is determined by an invariant of curves called the
  genus. In particular\, if $C_2$ has genus $g(C_2)\\geq 2$ (i.e.\, $C_2$ i
 s hyperbolic)\, then there are only finitely many non-constant morphisms $
 C_1 \\to C_2$ where $C_1$ is any curve\, and moreover\, the degree of any 
 map $C_1 \\to C_2$ is bounded linearly in $g(C_1)$ by the Riemann--Hurwitz
  formula. \n\nIn this talk\, I will explain the above story and discuss a 
 higher dimensional generalization of this result. To this end\, I will des
 cribe the conjectures of Demailly and Lang which predict a relationship be
 tween the geometry of varieties\, topological properties of Hom-schemes\, 
 and the behavior of rational points on varieties. To conclude\, I will ske
 tch a proof of a variant of these conjectures\, which roughly says that if
  $X/K$ is a hyperbolic variety\, then for every smooth projective curve $C
 /K$ of genus $g(C)\\geq 0$\, the degree of any map $C\\to X$ is bounded un
 iformly in $g(C)$.\n\nJoin Zoom Meeting\nhttps://ubc.zoom.us/j/67843190638
 ?pwd=eUJsc1oyY2xhYnM4NmU3OW1sTEV2dz09\n\nMeeting ID: 678 4319 0638\nPassco
 de: 999070\n
LOCATION:https://researchseminars.org/talk/UBC_NTS/1/
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