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Fix a vector of integers A = (a_1,a_2,...,a_n). The double ramification cycle DR_{g,A}
is formally defined via the virtual fundamental class of the space of relative stable
maps to the projective line, and informally is the locus in the moduli space of
n-pointed stable curves parametrizing curves on which the line bundle O(\sum a_ix_i)
is trivial. One of the great achievements of the field was a calculation of this cycle
in the tautological ring by Janda, Pandharipande, Pixton and Zvonkine. The methods of
JPPZ have however been limited to the DR, and have not been sufficient to understand
related cycles -- the Brill-Noether cycles w_{g,r,A}^d, which roughly speaking
paramatrize curves on which O(\sum a_ix_i) has r+1 linearly independent sections, and
the higher ramification cycles, which arise from the virtual fundamental class of the
space of relative stable maps to higher dimensional toric varieties.
In this talk, I will discuss how recent intersection-theoretic techniques originating from logarithmic and tropical geometry, and a logarithmic study of compactified Jacobians are the common framework underlying all these problems, and in particular, recover the calculation of the DR, but also lead to explicit formulas for the Brill-Noether and higher ramification cycles as well. |