Segre numbers of tautological bundles on Hilbert schemes of surfaces
--  Claire Voisin, March 29, 2019

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The top Segre classes of tautological bundles of punctual  
Hilbert schemes of  surfaces are numbers which  generalize the degrees  
of secant varieties. They can be assembled  in a  generating function  
and  The Lehn conjecture gives a complete description of this  
function, as obtained from a rational function by a change of  
variables.  We establish geometric vanishings of these top Segre  
classes in certain ranges  for  K3 surfaces (which had been first  
obtained by Marian-Oprea-Pandharipande  by different methods) and also  
for K3 surfaces blown-up at one point. We show how all the Segre  
numbers  (hence the whole generating series) are formally determined  
by these vanishings and we thus reduce the Lehn conjecture  to showing  
that the Lehn function also has these vanishing properties.  
Marian-Oprea-Pandharipande and Szenes-Vergne in turn used the present  
results to complete  the proof of the Lehn  conjecture.