Collectanea Mathematica最新文献

A marked Prym curve is a triple ((C,alpha ,T_d)) where C is a smooth algebraic curve, (alpha ) is a (2-)torsion line bundle on C, and (T_d) is a divisor of degree d. We give obstructions—in terms of Gaussian maps—for a marked Prym curve ((C,alpha ,T_d)) to admit a singular model lying on an Enriques surface with only one ordinary singular point of multiplicity d, such that (T_d) is the pull-back of the singular point by the normalization map. More precisely, let (S, H) be a polarized Enriques surface and let (C, f) be a smooth curve together with a morphism (f:C rightarrow S) birational onto its image and such that (f(C) in |H|), f(C) has exactly one ordinary singular point of multiplicity d. Let (alpha =f^*omega _S) and (T_d) be the divisor over the singular point of f(C). We show that if H is sufficiently positive then certain natural Gaussian maps on C, associated with (omega _C), (alpha ), and (T_d) are not surjective. On the contrary, we show that for the general triple in the moduli space of marked Prym curves ((C,alpha ,T_d)), the same Gaussian maps are surjective.

Abstract

Let I be a perfect ideal of height two in (R=k[x_1, ldots , x_d]) and let (varphi ) denote its Hilbert–Burch matrix. When (varphi ) has linear entries, the algebraic structure of the Rees algebra ({mathcal {R}}(I)) is well-understood under the additional assumption that the minimal number of generators of I is bounded locally up to codimension (d-1) . In the first part of this article, we determine the defining ideal of ({mathcal {R}}(I)) under the weaker assumption that such condition holds only up to codimension (d-2) , generalizing previous work of P. H. L. Nguyen. In the second part, we use generic Bourbaki ideals to extend our findings to Rees algebras of linearly presented modules of projective dimension one.

Let ((mathcal {G},otimes )) be any closed symmetric monoidal Grothendieck category. We show that K-flat covers exist universally in the category of chain complexes and that the Verdier quotient of (K(mathcal {G})) by the K-flat complexes is always a well generated triangulated category. Under the further assumption that (mathcal {G}) has a set of (otimes)-flat generators we can show more: (i) The category is in recollement with the (otimes)-pure derived category and the usual derived category, and (ii) The usual derived category is the homotopy category of a cofibrantly generated and monoidal model structure whose cofibrant objects are precisely the K-flat complexes. We also give a condition guaranteeing that the right orthogonal to K-flat is precisely the acyclic complexes of (otimes)-pure injectives. We show this condition holds for quasi-coherent sheaves over a quasi-compact and semiseparated scheme.

We prove the Kato–Ponce inequality (fractional normed Leibniz rule) for multiple factors in the setting of multiple weights ((A_{vec P}) weights). This improves existing results to the product of m factors and extends the class of known weights for which the inequality holds.

In this paper we introduce the atomic Hardy space (mathcal {H}^1((0,infty ),gamma _alpha )) associated with the non-doubling probability measure (dgamma _alpha (x)=frac{2x^{2alpha +1}}{Gamma (alpha +1)}e^{-x^2}dx) on ((0,infty )), for ({alpha >-frac{1}{2}}). We obtain characterizations of (mathcal {H}^1((0,infty ),gamma _alpha )) by using two local maximal functions. We also prove that the truncated maximal function defined through the heat semigroup generated by the Laguerre differential operator is bounded from (mathcal {H}^1((0,infty ),gamma _alpha )) into (L^1((0,infty ),gamma _alpha )).

In this article, we introduce a relation including ideals of an evolution algebra A and hereditary subsets of vertices of its associated graph and establish some properties among them. This relation allows us to determine maximal ideals and ideals having the absorption property of an evolution algebra in terms of its associated graph. In particular, the maximal ideals can be determined through maximal hereditary subsets of vertices except for those containing (A^2). We also define a couple of order-preserving maps, one from the sets of ideals of an evolution algebra to that of hereditary subsets of the corresponding graph, and the other in the reverse direction. Conveniently restricted to the set of absorption ideals and to the set of hereditary saturated subsets, this is a monotone Galois connection. According to the graph, we characterize arbitrary dimensional finitely-generated (as algebras) evolution algebras under certain restrictions of its graph. Furthermore, the simplicity of finitely-generated perfect evolution algebras is described on the basis of the simplicity of the graph.

This paper shows that on the Bergman space of the open unit disk, the Toeplitz operator (T_{{overline{p}}+varphi }) and the small Hankel operator (Gamma _psi) commute only in the obvious cases, where (varphi) and (psi) are both bounded analytic functions, and p is an analytic polynomial.

Abstract

An investigation is made of the generalized Cesàro operators (C_t) , for (tin [0,1]) , when they act on the space (H({{mathbb {D}}})) of holomorphic functions on the open unit disc ({{mathbb {D}}}) , on the Banach space (H^infty ) of bounded analytic functions and on the weighted Banach spaces (H_v^infty ) and (H_v^0) with their sup-norms. Of particular interest are the continuity, compactness, spectrum and point spectrum of (C_t) as well as their linear dynamics and mean ergodicity.

We prove new results on the interplay between reduction numbers and the Castelnuovo–Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff–Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we largely generalize a result of A. Mafi on ideals in two-dimensional Cohen–Macaulay local rings, by extending it to arbitrary dimension (and allowing for the setting relative to a Cohen–Macaulay module). We derive a number of applications, including a characterization of (polynomial) ideals of linear type, progress on the theory of generalized Ulrich ideals, and improvements of results by other authors.

We provide the necessary and sufficient condition for a pointwise slant submanifold with respect to two anti-commuting almost Hermitian structures to be also pointwise slant with respect to a family of almost Hermitian structures generated by them. On the other hand, we show that the property of being pointwise slant is transitive on a class of proper pointwise slant immersed submanifolds of almost Hermitian manifolds. We illustrate the results with suitable examples.