Abstract. Given a squarefree monomial ideal I ⊆ R = k[x 1 , . . . , x n ], we show that α(I), the Waldschmidt constant of I, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of I. By applying results from fractional graph theory, we can then express α(I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing α(I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on α(I), thus verifying a conjecture of Cooper-Embree-Hà-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of P n with few components compared to n, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.
In a recently published paper [Trans. Amer. Math. Soc. 363 (2011) 229-257], Migliore, Miró-Roig and Nagel show that the weak Lefschetz property (WLP) can fail for an ideal I ⊆ K[x 1, . . . , x4] generated by powers of linear forms. This is in contrast to the analogous situation in K[x 1, x2, x3], where WLP always holds [H. Schenck and A. Seceleanu, Proc. Amer. Math. Soc. 138 (2010) 2335-2339]. We use the inverse system dictionary to connect I to an ideal of fat points, and show that the failure of WLP for powers of linear forms is connected to the geometry of the associated fat point scheme. Recent results of Sturmfels and Xu [J. Eur. Math. Soc. 12 (2010) 429-459] allow us to relate the WLP to Gelfand-Tsetlin patterns.
Abstract. We show that an Artinian quotient of an ideal I ⊆ K[x, y, z] generated by powers of linear forms has the Weak Lefschetz Property. If the syzygy bundle of I is semistable, the property follows from results of Brenner-Kaid. Our proof works without this hypothesis, which typically does not hold.
Abstract. Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [BH1, Cu, ELS, HaHu, HoHu, Hu1, Hu2] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [BH1, GHvT].There have been exciting new developments in this area recently. It had been expected for several years that I Nr−N+1 ⊆ I r should hold for the ideal I of any finite set of points in P N for all r > 0, but in the last year various counterexamples have now been constructed (see [DST, HS, C. et al]), all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.
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