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We prove that $$\omega \Delta ^{\prime}_{e_{k}}e_{n}|_{t=0}$$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $$\Delta $$-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered tableau. Our proof is geometric, and shows that the $$\Delta $$-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $$t$$ and $$t^{2}$$ coefficients of $$\omega \Delta ^{\prime}_{e_{k}}e_{n}$$. 

We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d morphisms from a general genus-g, n-marked curve C to Pr, sending the marked points on C to specified general points in Pr, is equal to (r + 1)g for sufficiently large d. This computation may be rephrased as an intersection problem on Grassmannians, which has a combinatorial interpretation in terms of Young tableaux by the Littlewood-Richardson rule. We give a bijection, generalizing the RSK correspondence, between the tableaux in question and the (r+1)ary sequences of length g, and we explore our bijection’s combinatorial properties. We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which r = 1 and several marked points map to the same point in P1, the number of morphisms is still 2g for sufficiently large d. 

We conjecture a simple combinatorial formula for the Schur expansion of the Frobenius series of the Sn-modules Rn,λ,s, which appear as the cohomology rings of the “∆-Springer” varieties. These modules interpolate between the Garsia-Procesi modules Rµ (which are the type A Springer fiber cohomology rings) and the rings Rn,k defined by Haglund, Rhoades, and Shimozono in the context of the Delta Conjecture. Our formula directly generalizes the known cocharge formula for Garsia-Procesi modules and gives a new cocharge formula for the Delta Conjecture at t = 0, by introducing battery-powered tableaux that “store” extra charge in their battery. Our conjecture has been verified by computer for all n ≤ 10 and s ≤ ℓ(λ)+2, as well as for n ≤ 8 and s ≤ ℓ(λ)+7. We prove it holds for several infinite families of n,λ,s.