Collectanea Mathematica最新文献

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematic areas. Grassmannians are the building blocks for skewsymmetric tensors. Although they are ubiquitous in the literature, the geometry of their secant varieties is not completely understood. In this work we determine the singular locus of the secant variety of lines to a Grassmannian Gr(k, V) using its structure as ({{,textrm{SL},}}(V))-variety. We solve the problems of identifiability and tangential-identifiability of points in the secant variety: as a consequence, we also determine the second Terracini locus to a Grassmannian.

We recall and delve into the different characterizations of the depth of an affine semigroup ring, providing an original characterization of depth two in three and four dimensional cases which are closely related to the existence of a maximal element in certain Apéry sets.

We describe Lorentzian manifolds that admit metric connections with parallel torsion having zero twistorial component and non-zero vectorial component. We also describe Lorentzian manifolds admitting metric connections with closed parallel skew-symmetric torsion.

The classification of compact homogeneous spaces of the form (M=G/K), where G is a non-simple Lie group, such that the standard metric is Einstein is still open. The only known examples are 4 infinite families and 3 isolated spaces found by Nikonorov and Rodionov in the 90 s. In this paper, we prove that most of these standard Einstein metrics are unstable as critical points of the scalar curvature functional on the manifold of all unit volume G-invariant metrics on M, providing a lower bound for the coindex in the case of Ledger–Obata spaces. On the other hand, examples of stable (in particular, local maxima) invariant Einstein metrics on certain homogeneous spaces of non-simple Lie groups are also given.

This paper focuses on a length-preserving flow for star-shaped curves with respect to the origin. Under the length-preserving flow, the evolving curve keeps star-shapedness and converges smoothly to a circle, which can be regarded as a Grayson-type theorem for star-shaped curves under this flow.

Let (X^{1,n}_r) be the blow-up of (mathbb {P}^1times mathbb {P}^n) in r general points. We describe the Mori cone of (X^{1,n}_r) for (rle n+2) and for (r = n+3) when (nle 4). Furthermore, we prove that (X^{1,n}_{n+1}) is log Fano and give an explicit presentation for its Cox ring.

We prove that (frac{Q}{Q-2}) is the Fujita exponent for a semilinear heat equation on an arbitrary stratified Lie group with homogeneous dimension Q. This covers the Euclidean case and gives new insight into proof techniques on nilpotent Lie groups. The equation we study has a forcing term which depends only upon the group elements and has positive integral. The stratified Lie group structure plays an important role in our proofs, along with test function method and Banach fixed point theorem.

The authors of Cao and Yan (J Differ Equ 251:1389–1414, 2011) have considered the following semilinear critical Neumann problem

where (varvec{Omega }) is a bounded domain in (varvec{mathbb {R}^{N}}) satisfying some geometric conditions, (varvec{nu }) is the outward unit normal of (varvec{partial Omega , 2^{*}:=frac{2 N}{N-2}}) and (varvec{g(t):=mu vert tvert ^{p-2} t-t,}) where (varvec{p in left( 2,2^{*}right) }) and (varvec{mu >0}) are constants. They proved the existence of infinitely many solutions with positive energy for the above problem if (varvec{N>max left( frac{2(p+1)}{p-1}, 4right) .}) In this present paper, we consider the case where the exponent (varvec{p in left( 1,2right) }) and we show that if (varvec{N>frac{2(p+1)}{p-1},}) then the above problem admits an infinite set of solutions with positive energy. Our main result extend that obtained by P. Han in [9] for the case of elliptic problem with Dirichlet boundary conditions.

We characterize the boundedness properties on the spaces (L^p( mathbb {H}^2)) of the maximal operator (M_mathcal {B}) where (mathcal {B}) is an arbitrary family of hyperbolic triangles stable by isometries.

Let R be a commutative noetherian ring and I an ideal of R. Assume that for all integers i the local cohomology module ({text {H}}_I^i(R)) is I-cofinite. Suppose that (R_mathfrak {p}) is a regular local ring for all prime ideals (mathfrak {p}) that do not contain I. In this paper, we prove that if the I-cofinite modules form an abelian category, then for all finitely generated R-modules M and all integers i, the local cohomology module ({text {H}}_I^i(M)) is I-cofinite.