On the geometry of Abel maps for nodal curves

In this paper we give local conditions to the existence of Abel maps for nodal curves that are limits of Abel maps for smooth curves. We use this result to construct Abel maps for any degree for nodal curves with two components.Comment: 30 pages, 1 figur

Enriched curves and their tropical counterpart

In her Ph.D. thesis, Main\`o introduced the notion of enriched structure on stable curves and constructed their moduli space. In this paper we give a tropical notion of enriched structure on tropical curves and construct a moduli space parametrizing these objects. Moreover, we use this construction to give a toric description of the scheme parametrizing enriched structures on a fixed stable curve.Comment: 41 pages, 13 figures; final version, to appear in Ann. Inst. Fourie

Splitting the cohomology of Hessenberg varieties and e-positivity of chromatic symmetric functions

For each indifference graph, there is an associated regular semisimple Hessenberg variety, whose cohomology recovers the chromatic symmetric function of the graph. The decomposition theorem applied to the forgetful map from the regular semisimple Hessenberg variety to the projective space describes the cohomology of the Hessenberg variety as a sum of smaller pieces. We give a combinatorial description of the Frobenius character of each piece. This provides a generalization of the symmetric functions attached to Stanley's local h-polynomials of the permutahedral variety to any Hessenberg variety. As a consequence, we can prove that the coefficient of $e_{\lambda}$, where $\lambda$ is any partition of length 2, in the e-expansion of the chromatic symmetric function of any indifference graph is non-negative.Comment: 23 page

A Torelli theorem for graphs via quasistable divisors

The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine compactified Jacobian. The fine compactified Jacobian of a curve comes with a natural stratification that can be regarded as a poset. Furthermore, this poset is entirely determined by the dual graph of the curve and is referred to as the poset of quasistable divisors on the graph. We present a combinatorial version of the Torelli theorem, which demonstrates that the poset of quasistable divisors of a graph completely determines the biconnected components of the graph (up to contracting separating edges). Moreover, we achieve a natural extension of this theorem to tropical curves.Comment: MSC: 05Cxx, 14Hxx. 22 page

Parabolic Lusztig varieties and chromatic symmetric functions

The characters of Kazhdan--Lusztig elements of the Hecke algebra over $S_n$ (and in particular, the chromatic symmetric function of indifference graphs) are completely encoded in the (intersection) cohomology of certain subvarieties of the flag variety. Considering the forgetful map to some partial flag variety, the decomposition theorem tells us that this cohomology splits as a sum of intersection cohomology groups with coefficients in some local systems of subvarieties of the partial flag variety. We prove that these local systems correspond to representations of subgroups of $S_n$. An explicit characterization of such representations would provide a recursive formula for the computation of such characters/chromatic symmetric functions, which could settle Haiman's conjecture about the positivity of the monomial characters of Kazhdan--Lusztig elements and Stanley--Stembridge conjecture about $e$-positivity of chromatic symmetric function of indifference graphs. We also find a connection between the character of certain homology groups of subvarieties of the partial flag varieties and the Grojnowski--Haiman hybrid basis of the Hecke algebra.Comment: 36 page

Wall-crossing of universal Brill-Noether classes

We give an explicit graph formula, in terms of decorated boundary strata classes, for the wall-crossing of universal Brill-Noether classes. More precisely, fix $n>0$ and $d<g$ , and two stability conditions $\phi^-, \phi^+$ for degree~$d$ compactified universal (over $\overline{\mathcal{M}}_{g,n}$) Jacobians that lie on opposite sides of a stability hyperplane. Our main result is a formula for the difference between the Brill-Noether classes, compared via the pullback along the (rational) identity map $\mathsf{Id} \colon \overline{\mathcal{J}}^d_{g,n} (\phi^+) \dashrightarrow \overline{\mathcal{J}}^d_{g,n} (\phi^-)$. The calculation involves constructing a resolution of the identity map by means of subsequent blow-ups.Comment: 66 pages, 1 figure. Comments are welcom