Rennemo, Jørgen Vold Homology of Hilbert schemes of points on a locally planar curve. (English) Zbl 1409.14011 J. Eur. Math. Soc. (JEMS) 20, No. 7, 1629-1654 (2018). Let \(C\) be an integral proper complex curve with compactified Jacobian \(J\). Letting \(C^{[n]}\) denote the Hilbert scheme of length \(n\) subschemes of \(C\), the Abel-Jacobi morphism \(\varphi: C^{[n]} \to J\) sends a closed subscheme \(Z\) to \({\mathcal I}_Z \otimes {\mathcal O}(x)^{\otimes n}\), where \(x \in C\) is a nonsingular point. When \(C\) has at worst planar singularities, both \(C^{[n]}\) and \(J\) are integral schemes with local complete intersection singularities according to A. B. Altman et al. [in: Real and compl. Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 1–12 (1977; Zbl 0415.14014)] and J. Briancon et al. [Ann. Sci. Éc. Norm. Supér. (4) 14, 1–25 (1981; Zbl 0463.14001)]. Furthermore \(\varphi\) has the structure of a \(\mathbb P^{n-g}\)-bundle for \(n \geq 2g-1\) by work of A. B. Altman and S. L. Kleiman [Adv. Math. 35, 50–112 (1980; Zbl 0427.14015)] so that the rational homology group \(H_* (C^{[n]})\) is determined by \(H_* (J)\). Recent work of D. Maulik and Z. Yun [J. Reine Angew. Math. 694, 27–48 (2014; Zbl 1304.14036)] and L. Migliorini and V. Shende [J. Eur. Math. Soc. (JEMS) 15, No. 6, 2353–2367 (2013; Zbl 1303.14019)] endows \(H^* (J)\) with a certain perverse filtration \(P\) for which \(H^* (C^{[n]})\) can be recovered from the \(P\)-graded space \(\text{gr}_*^P H^* (J)\). Motivated by these results and a suggestion of Richard Thomas, the author shows how \(H_* (C^{[n]})\) can be recovered from a filtration on \(H_* (J)\) using a method not reliant on perverse sheaves. Taking an approach inspired by work of H. Nakajima [Ann. Math. (2) 145, No. 2, 379–388 (1997; Zbl 0915.14001)], he defines two pairs of creation and annihilation operators acting on \(V(C)=\bigoplus_{n \geq 0} H_* (C^{[n]})\). The first pair \(\mu_{\pm} [\text{pt}]\) corresponds to adding or removing a nonsingular point in \(C\). The second pair \(\mu_{\pm}[C]\) come from the respective projections \(p,q\) from the flag Hilbert scheme \(C^{[n,n+1]}\) to \(C^{[n]}\) and \(C^{[n+1]}\), namely \(q_* p^{!}\) and \(p_* q^{!}\) for appropriate Gysin maps \(p^!\) and \(q^!\). The main theorem states that the subalgebra of \(\text{End} (V(C))\) generated by \(\mu_{\pm} [\text{pt}], \mu_{pm}[C]\) is isomorphic to the Weyl algebra \(\mathbb Q [x_1, x_2, \partial_1, \partial_2]\) and that the natural map \(W \otimes \mathbb Q [\mu_+ [\text{pt}], \mu_+ [C]] \to V(C)\) is an isomorphism, where \(W\) is the intersection of the kernels of \(\mu_- [\text{pt}]\) and \(\mu_- [C]\); moreover the Abel-Jacobi pushforward map \(\varphi_*: V(C) \to H_* (J)\) induces an isomorphism \(W \cong H_* (J)\). Dual variations for cohomology groups recover and strengthen the results of Maulik-Yun [Zbl 1304.14036] and Migliorini-Shende [Zbl 1303.14019]. Reviewer: Scott Nollet (Fort Worth) Cited in 2 ReviewsCited in 5 Documents MSC: 14C05 Parametrization (Chow and Hilbert schemes) 14H40 Jacobians, Prym varieties 14H20 Singularities of curves, local rings Keywords:locally planar curves; Hilbert scheme; compactified Jacobian; Weyl algebra Citations:Zbl 0415.14014; Zbl 0463.14001; Zbl 0427.14015; Zbl 1304.14036; Zbl 1303.14019; Zbl 0915.14001 × Cite Format Result Cite Review PDF Full Text: DOI arXiv