We prove Demailly's Conjecture when m=2 for any set of very general points, any set of at least 2^N general points in ℙn and any set of general points in ℙ3,ℙ4,ℙ5 except four cases in ℙ5.   arxiv  journal

We completely characterize and give a full list of one dimensional complexes with vanishing T2. We compute the graded components of T2 of uniform matroid simplicial complex. We also show that for all matroids of corank at most 2, T2 vanishes, and conjecture that all connected matroids with vanishing T2 are of corank at most 2.   arxiv  journal

We obtain recursive formulae for regularity, a-invariant, and multiplicity of geometrically vertex decomposable ideals. We apply these results for several classes of gvd ideals including toric ideals of bipartite graphs.   arxiv  journal

We characterize the Noetherian property of the Rees algebra of a graded family of monomial ideals via its Newton-Okounkov body; and present a combinatorial interpretation for its analytic spread. A related result for Newton-Okounkov body of m-primary ideals is obtained. We also apply these results to bound the generation type and the Veronese degree of the symbolic Rees algebra of a monomial ideal.   arxiv  journal

We show stable Harbourne—Huneke Conjectures and Chudnovsky's Conjecture for ideals of any number of general points in ℙn.   arxiv  journal

We establish a duality in two important algebro-geometric contexts: one is between the sequence of initial degrees of symbolic powers of a radical ideal and the sequence of regularity of a quotient by ideals generated by powers of linear forms; the other one is between the multipoint Seshadri constant and the asymptotic regularity of a set of points in projective space.   arxiv  journal

We calculate the initial degree of almost all symbolic powers and Waldschmidt constant of Fermat-like ideals of planes arrangement in ℙ3 and of the lines arrangement corresponds to group A3 , and show that Harbourne-Huneke Conjectures for these ideals can be checked purely by these invariant and the maximal generating degrees.   arxiv  journal 

We calculate the initial degree of symbolic powers of Fermat ideals in all unknown cases and show that Harbourne-Huneke Conjectures for these ideals can be verified by these invariant and the maximal generating degrees. We also calculate the Waldschmidt constant and resurgence number in the unknown cases.   arxiv  journal

We present RandomRationalPoints, a package in Macaulay2 designed mainly to identify rational and geometric points in a variety over a finite field.  An application is to obtain non-vanishing minors of a given size in a given matrix, by evaluating the matrix at a point.   arxiv  journal

We introduce quasi f-ideal, which is a generalization of f-ideal, and investigate some basic properties and classification of them. We introduce quasi f-complexes and show that they are connected. We also give a construction of Cohen-Macaulay quasi f-graphs.   arxiv  journal

We show Chudnovsky's Conjecture for ideals of sufficiently many general points in ℙn, by showing the stable versions of the Harbourne—Huneke conjectures. We also verify Chudnovsky's Conjecture for ideals of fat points with general support and with uniform multiplicities at least 2.   arxiv  journal

We show each of Demailly's inequalities holds for ideals of sufficiently many general points in ℙn  by showing that the Harbourne-Huneke containment that would imply Demailly's bound holds for infinitely many values.  We also verify the containment for star configurations and generic determinantal ideals.   arxiv  journal

We develop the notions of Newton non-degenerate (NND) ideals and Newton polyhedra of ideals in regular local rings. We show that a Noetherian graded family of $\mm$-primary ideals in a regular local ring satisfies the "Multiplicity = Volume" formula if and only if it contains a subfamily of NND ideals.  arxiv

We show that the asymptotic regularity of a graded family exists when the family consists of artinian ideals, or Cohen-Macaulay ideals of the same codimension, or when its Rees algebra is Noetherian. We also show that the asymptotic regularity can be realized via vertex data of the associated Newton-Okounkov body in certain situations. We give a negative answer to the question of whether the limit of the regularity a family given by intersections or sums of ideals exist. These provide ample evidence showing that the asymptotic regularity of the family of symbolic powers of a homogeneous ideal may not exist.  arxiv

We relate the asymptotic resurgence number of a pair of graded families of ideals with combinatorial data of their associated convex bodies. The classes of ideals discussed include monomial ideals, and classical invariant ideals such as determinantal ideals or ideals of Pfaffians.  arxiv

We prove that the asymptotic depth of any Inc-invariant chain of edge ideals can attain only one of two possible values and provide the formula explicitly. We also compute the asymptotic homology of the independence complexes of those ideals.   arxiv

We show the asymptotic relationship between the limit of the normalized length function of a multi-p-family of ideals and that of its shifted family under a linear growth assumptions in a local domain of characteristic p. We obtain a generalized version of a formula due to Wantanabe-Yoshida for certain p-families and give an instance of the existence of a mixed multiplicity version of multi-p-families of ideals.   arxiv

We introduce and study the resurgence and asymptotic resurgence numbers associated to a pair of graded families of ideals in a Noetherian ring. These notions generalize the well-studied resurgence and asymptotic resurgence of an ideal in a polynomial ring, thus, provide a general framework to study (non)containment between ideals in graded families.   arxiv