Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all its powers are integrally closed).For a monomial ideal the skew Waldschmidt constants have an interpretation involving the symbolic polyhedron defined by Cooper, Embree, Hà, and Hoefel. Using this intuition we provide several examples of squarefree monomial ideals whose resurgence and asymptotic resurgence are different.

Let M and N be two monomials of the same degree, and let I be the smallest Borel ideal containing M and N . We show that the toric ring of I is Koszul by constructing a quadratic Gröbner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.

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We give a complete classification of free and non-free multiplicities on the A 3 braid arrangement. Namely, we show that all free multiplicities on A 3 fall into two families that have been identified by Abe-Terao-Wakefield (2007) and Abe-Nuida-Numata (2009). The main tool is a new homological obstruction to freeness derived via a connection to multivariate spline theory.

Abstract. We study the module C r (P) of piecewise polynomial functions of smoothness r on a pure n-dimensional polytopal complex P ⊂ R n , via an analysis of certain subcomplexes P W obtained from the intersection lattice of the interior codimension one faces of P. We obtain two main results: first, we show that the vector space C r d (P) of splines of degree ≤ d has a basis consisting of splines supported on the P W for d 0. We call such splines lattice-supported. This shows that an analog of the notion of a star-supported basis for C r d (∆) studied by Alfeld-Schumaker in the simplicial case holds [3]. Second, we provide a pair of conjectures, one involving lattice-supported splines, bounding how large d must be so that dim R C r d (P) agrees with the McDonald-Schenck formula [14]. A family of examples shows that the latter conjecture is tight. The proposed bounds generalize known and conjectured bounds in the simplicial case.

We define a chain complex for generalized splines on graphs, analogous to that introduced by Billera and refined by Schenck-Stillman for splines on polyhedral complexes. The hyperhomology of this chain complex yields bounds on the projective dimension of the ring of generalized splines. We apply this construction to the module of derivations of a graphic multi-arrangement, yielding homological criteria for bounding its projective dimension and determining freeness. As an application, we show that a graphic arrangement admits a free constant multiplicity iff it splits as a product of braid arrangements.1991 Mathematics Subject Classification. Primary 13P20, Secondary 13D02, 32S22, 05E40.