Beyond the tautological ring  

Rahul Pandharipande

5 July 2023


(0) The study of R^*(M_g), R^*(\barM_g,n) is well-developed. You can prove
    Pixton's relation are a complete set or you can disprove Pixton's conjecture.
    Both are hard to do.


    But now there are at least 3 new directions to go instead:


(1) The Hurwitz extensions: add to the tautological rings the closure under all
    correspondences defined by the fundamental classes of Hurwitz moduli spaces

    G-Hur^g_h(mu^1,...mu^n)

    of degree d covers with monodromy group G in S_d.

    See papers of Lian, Schmitt-van Zelm (restricted mainly to the case of
    Galois covers).

    No definitive general results yet except for the statement that the
    Hurwitz extension is actually bigger than R^*(\barM_g,n) by Graber-P, van Zelm.


    Joke:

      Q: Why are movies about abelian groups dull?
      A: Because all the characters are 1-dimensional.


    I mentioned that the GW theory of orbifold P1 provides nontrivial relations
    in the subalgebra generated by Hurwitz covers with groups G=Z/nZ by localization.
    In spite of the joke, I think this will be an interesting movie.


(2) Direction of Canning-Larson-Payne: start adding more classes by genus.
    Their study is in cohomology (so subalgebras of H*(\barM_g,n)).

    For example: add the entire cohomology of \barM_{1,n}.
 
    How to do this? You can enrich the additive generators in my paper with Graber
    by giving yourself the option on a genus 1 vertex of a stable graph of
    adding *any* cohomology class. There is some control here since by Getzler
    and Petersen, we know all the cohomology in genus 1.

    We obtain a ring RH^*(\barM_g,n)^1 in H*(\barM_g,n).

    The kind of results we could hope for are of following form:

         RH^k(\barM_g,n)^1= H^k(\barM_g,n) for k<=15.

    A lot, but not all, of the above statement has been proven by
    Canning-Larson-Payne.

      [Canning explains that homology is easier and

       RH_k(\barM_g,n)^1= H_k(\barM_g,n) for k<=14

       is a Theorem of Canning-Larson-Payne. The cohomology
       result is expected to be true, but harder to prove.]

    Then we can add more genera. For example

        RH^*(\barM_g,n)^h in H^*(\barM_g,n)

    allows decoration of all vertices of genus <=h by any class.

    Question (by Lian): Are the classes of the bi-elliptic loci
    considered in Graber-P in RH^*(\barM_2,n)^1?

    Answer in the lecture: Not sure, but maybe Petersen knows.

      [The answer is No by the following argument.
       The bi-elliptic class of Graber-P in Ch^11(\barM_2,20)
       which pulls-back to the diagonal of \barM_1,11 x \barM_1,11
       cant be in RH^22(\barM_2,20)^1 since there is nothing there
       which has Kunneth components in H^11,0 \tensor H^0,11
       after pull-back to \barM_1,11 x \barM_1,11. Because to have these
       interesting Kunneth components, we need to place the H^11,0 and H^0,11
       on some elliptic components. So we need a node together with these
       classes which is then in RH^>22(\barM_2,20)^1.

       Canning gave a different argument -- also with answer No.]
   
    An interesting variant of (1)+(2): consider the subsystem of
    H^*(\barM_g,n) generated by the tautological classes, correspondences
    defined by Hurwitz covers G-Hur^g_1 (with bottom curve of genus 1), and
    all genus 1 cohomology in the sense of Canning-Larson-Payne.

    But what is the result to prove? I guess the first question could be
    to determine how much bigger this is than RH^*(\barM_2,n)^1,
    an extension of Lian's question.


(3) Log Chow theory.


    I motivated the study of Log GW by a few results (the GW/Kronecker correspondence
    in my old work with Gross and Siebert, Abramovich-Wise birational invariance,
    Kumaran-DR about toric log GW in terms of logDR, the log product formula of Herr and DR,
    the logDR paper.

    Finally, someone asked what does DR stand for?

    Answer in the lecture: Dhruv Ranganthan or Double Ramification depending upon context.

    I defined LogCH for \barM_g,n and stated two results in an upcoming
    paper with Schmitt and DR:

     Theorem I.  LogCH(\barM_0,n) = PP(ConeComplex) / WDVV.

     Theorem II. Statement that full knowledge of R^*(\Mbar_g,n) determines
                 in principle full knowledge of logR^*(\Mbar_g,n).

    There are two proofs of Theorem I, but I sketched the more elegant path
    using the relationship to Chow quotients, toric embeddings, and wonderful
    compactifications.

    The simplicity of Theorem I cant be fully understood until you see
    how completely the parallel statements in g=1 go wrong.

    Pixton's DR relations are lifted to logR^*(\Mbar_g,n) by the logDR
    theory (Holmes, Molcho, P, Pixton, Schmitt), but how to lift Pixton's
    3-spin relations is an open question.