48 research outputs found
Geometric lower bounds for generalized ranks
We revisit a geometric lower bound for Waring rank of polynomials (symmetric
rank of symmetric tensors) of Landsberg and Teitler and generalize it to a
lower bound for rank with respect to arbitrary varieties, improving the bound
given by the "non-Abelian" catalecticants recently introduced by Landsberg and
Ottaviani. This is applied to give lower bounds for ranks of multihomogeneous
polynomials (partially symmetric tensors); a special case is the simultaneous
Waring decomposition problem for a linear system of polynomials. We generalize
the classical Apolarity Lemma to multihomogeneous polynomials and give some
more general statements. Finally we revisit the lower bound of Ranestad and
Schreyer, and again generalize it to multihomogeneous polynomials and some more
general settings.Comment: 43 pages. v2: minor change
Bounding symbolic powers via asymptotic multiplier ideals
We revisit a bound on symbolic powers found by Ein-Lazarsfeld-Smith and
subsequently improved by Takagi-Yoshida. We show that the original argument of
Ein-Lazarsfeld-Smith actually gives the same improvement. On the other hand, we
show by examples that any further improvement based on the same technique
appears unlikely. This is primarily an exposition; only some examples and
remarks might be new.Comment: 10 pages. Primarily exposition. Originally written as appendix to
lecture notes by Brian Harbourne. v2: Minor changes. v3: Final version,
appeared in Ann. Univ. Pedagog. Crac. Stud. Mat
Castelnuovo-Mumford regularity and arithmetic Cohen-Macaulayness of complete bipartite subspace arrangements
We give the Castelnuovo-Mumford regularity of arrangements of (n-2)-planes in
P^n whose incidence graph is a sufficiently large complete bipartite graph, and
determine when such arrangements are arithmetically Cohen-Macaulay.Comment: v3: Minor changes, 5p
Topological criteria for schlichtness
We give two sufficient criteria for schlichtness of envelopes of holomorphy
in terms of topology. These are weakened converses of results of Kerner and
Royden. Our first criterion generalizes a result of Hammond in dimension 2.
Along the way we also prove a generalization of Royden's theorem.Comment: v2: 3 pages; added new results, including generalization of a theorem
of Royde
Lower bound for ranks of invariant forms
We give a lower bound for the Waring rank and cactus rank of forms that are
invariant under an action of a connected algebraic group. We use this to
improve the Ranestad--Schreyer--Shafiei lower bounds for the Waring ranks and
cactus ranks of determinants of generic matrices, Pfaffians of generic
skew-symmetric matrices, and determinants of generic symmetric matrices.Comment: 13 page