Abstract. Let n be a fixed positive integer and h : {1, 2, . . . , n} → {1, 2, . . . , n} a Hessenberg function. The main results of this paper are twofold. First, we give a systematic method, depending in a simple manner on the Hessenberg function h, for producing an explicit presentation by generators and relations of the cohomology ring H * (Hess(N, h)) with Q coefficients of the corresponding regular nilpotent Hessenberg variety Hess(N, h). Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring H * (Hess(N, h)) of the regular nilpotent Hessenberg variety and the Sn-invariant subring H * (Hess(S, h)) Sn of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the Sn-action on H * (Hess(S, h)) defined by Tymoczko). Our second main result implies that dim Q H k (Hess(N, h)) = dim Q H k (Hess(S, h)) Sn for all k and hence partially proves the Shareshian-Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley-Stembridge conjecture. A proof of the full Shareshian-Wachs conjecture was recently given by Brosnan and Chow, but in our special case, our methods yield a stronger result (i.e. an isomorphism of rings) by more elementary considerations. This paper provides detailed proofs of results we recorded previously in a research announcement.

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In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton-Okounkov bodies of the two-dimensional Peterson variety.