1,735 research outputs found
A note on the multiplicity of determinantal ideals
Herzog, Huneke, and Srinivasan have conjectured that for any homogeneous
-algebra, the multiplicity is bounded above by a function of the maximal
degrees of the syzygies and below by a function of the minimal degrees of the
syzygies. The goal of this paper is to establish the multiplicity conjecture of
Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay
algebras over a field for -algebras being a
determinantal ideal of arbitrary codimension
The renormalized Hamiltonian truncation method in the large expansion
Hamiltonian Truncation Methods are a useful numerical tool to study strongly
coupled QFTs. In this work we present a new method to compute the exact
corrections, at any order, in the Hamiltonian Truncation approach presented by
Rychkov et al. in Refs. [1-3]. The method is general but as an example we
calculate the exact and some of the contributions for the
theory in two dimensions. The coefficients of the local expansion calculated in
Ref. [1] are shown to be given by phase space integrals. In addition we find
new approximations to speed up the numerical calculations and implement them to
compute the lowest energy levels at strong coupling. A simple diagrammatic
representation of the corrections and various tests are also introduced.Comment: JHEP version, typos fixed in Appendix and eq. (23
Ideals generated by submaximal minors
The goal of this paper is to study irreducible families W(b;a) of codimension
4, arithmetically Gorenstein schemes X of P^n defined by the submaximal minors
of a t x t matrix A with entries homogeneous forms of degree a_j-b_i. Under
some numerical assumption on a_j and b_i we prove that the closure of W(b;a) is
an irreducible component of Hilb^{p(x)}(P^n), we show that Hilb^{p(x)}(P^n) is
generically smooth along W(b;a) and we compute the dimension of W(b;a) in terms
of a_j and b_i. To achieve these results we first prove that X is determined by
a regular section of the twisted conormal sheaf I_Y/I^2_Y(s) where
s=deg(det(A)) and Y is a codimension 2, arithmetically Cohen-Macaulay scheme of
P^n defined by the maximal minors of the matrix obtained deleting a suitable
row of A.Comment: 22 page
On the shape of a pure O-sequence
An order ideal is a finite poset X of (monic) monomials such that, whenever M
is in X and N divides M, then N is in X. If all, say t, maximal monomials of X
have the same degree, then X is pure (of type t). A pure O-sequence is the
vector, h=(1,h_1,...,h_e), counting the monomials of X in each degree.
Equivalently, in the language of commutative algebra, pure O-sequences are the
h-vectors of monomial Artinian level algebras. Pure O-sequences had their
origin in one of Richard Stanley's early works in this area, and have since
played a significant role in at least three disciplines: the study of
simplicial complexes and their f-vectors, level algebras, and matroids. This
monograph is intended to be the first systematic study of the theory of pure
O-sequences. Our work, making an extensive use of algebraic and combinatorial
techniques, includes: (i) A characterization of the first half of a pure
O-sequence, which gives the exact converse to an algebraic g-theorem of Hausel;
(ii) A study of (the failing of) the unimodality property; (iii) The problem of
enumerating pure O-sequences, including a proof that almost all O-sequences are
pure, and the asymptotic enumeration of socle degree 3 pure O-sequences of type
t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents
perhaps the strongest possible structural result short of an (impossible?)
characterization; (v) A pithy connection of the ICP with Stanley's matroid
h-vector conjecture; (vi) A specific study of pure O-sequences of type 2,
including a proof of the Weak Lefschetz Property in codimension 3 in
characteristic zero. As a corollary, pure O-sequences of codimension 3 and type
2 are unimodal (over any field); (vii) An analysis of the extent to which the
Weak and Strong Lefschetz Properties can fail for monomial algebras; (viii)
Some observations about pure f-vectors, an important special case of pure
O-sequences.Comment: iii + 77 pages monograph, to appear as an AMS Memoir. Several, mostly
minor revisions with respect to last year's versio
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