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Abstract: Abstract This is a study of the sequences of Betti numbers of finitely generated modules over a complete intersection local ring, R. The subsequences \((\beta ^R_i(M))\) with even, respectively, odd i are known to be eventually given by polynomials in i with equal leading terms. We show that these polynomials coincide if \({{I}{}^{\scriptscriptstyle \square }}\) , the ideal generated by the quadratic relations of the associated graded ring of R, satisfies \({\text {height}}{{I}{}^{\scriptscriptstyle \square }} \ge {\text {codim}}R -1\) , and that the converse holds if R is homogeneous or \({\text {codim}}R \le 4\) . Subsequently Avramov, Packauskas, and Walker proved that the terms of degree \(j > {\text {codim}}R -{\text {height}}{{I}{}^{\scriptscriptstyle \square }}\) of the even and odd Betti polynomials are equal. We give a new proof of that result, based on an intrinsic characterization of residue rings of c.i. local rings of minimal multiplicity obtained in this paper. We also show that that bound is optimal. PubDate: 2024-08-04 DOI: 10.1007/s13348-024-00449-5
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Abstract: Abstract In this paper, we introduce a class of infinite Lie conformal algebras \({\mathfrak {B}}(\alpha ,\beta ,p)\) , which are the semi-direct sums of Block type Lie conformal algebra \({\mathfrak {B}}(p)\) and its non-trivial conformal modules of \({\mathbb {Z}}\) -graded free intermediate series. The annihilation algebras are a class of infinite-dimensional Lie algebras, which include a lot of interesting subalgebras: Virasoro algebra, Block type Lie algebra, twisted Heisenberg–Virasoro algebra and so on. We give a complete classification of all finite non-trivial irreducible conformal modules of \({\mathfrak {B}}(\alpha ,\beta ,p)\) for \(\alpha ,\beta \in {\mathbb {C}}, p\in {\mathbb {C}}^*\) . As an application, the classifications of finite irreducible conformal modules over a series of finite Lie conformal algebras \({\mathfrak {b}}(n)\) for \(n\ge 1\) are given. PubDate: 2024-06-24 DOI: 10.1007/s13348-024-00448-6
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Abstract: Abstract In this paper we show the existence of cones over a 8-dimensional rational sphere at the boundary of the Mori cone of the blow-up of the plane at \(s\ge 13\) very general points. This gives evidence for De Fernex’s strong \(\Delta \) -conjecture, which is known to imply Nagata’s conjecture. This also implies the existence of a multitude of good and wonderful rays as defined in Ciliberto et al. (Clay Math Proc 18:185–203, 2013). PubDate: 2024-06-18 DOI: 10.1007/s13348-024-00447-7
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Abstract: Abstract This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces exhibit. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese and Del Pezzo surfaces. The main difficulties and the possible approaches to the case of general surfaces are discussed for and complemented by the example of Bordiga surfaces. PubDate: 2024-05-29 DOI: 10.1007/s13348-024-00444-w
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Abstract: Abstract We consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo surfaces with torus action of Picard number one up to Picard index \( 10\,000 \) . PubDate: 2024-05-23 DOI: 10.1007/s13348-024-00443-x
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Abstract: The paper contains the study of the weighted \(L^{p_1}\times L^{p_2}\times \ldots \times L^{p_m}\rightarrow L^p\) estimates for the multilinear maximal operator, in the context of abstract probability spaces equipped with a tree-like structure. Using the Bellman function method, we identify the associated optimal constants in the symmetric case \(p_1=p_2=\ldots =p_m\) , and a tight constant for remaining choices of exponents. PubDate: 2024-05-01 DOI: 10.1007/s13348-022-00390-5
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Abstract: Abstract We study stable trace ideals in one dimensional local Cohen–Macaulay rings and give numerous applications. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00391-y
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Abstract: Abstract This paper is devoted to a deep analysis of the process known as Cheeger deformation, applied to manifolds with isometric group actions. Here, we provide new curvature estimates near singular orbits and present several applications. As the main result, we answer a question raised by a seminal result of Searle–Wilhelm about lifting positive Ricci curvature from the quotient of an isometric action. To answer this question, we develop techniques that can be used to provide a substantially streamlined version of a classical result of Lawson and Yau, generalize a curvature condition of Chavéz, Derdzinski, and Rigas, as well as, give an alternative proof of a result of Grove and Ziller. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00396-7
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Abstract: Abstract Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to k-planes with the restriction that their intersection has a prescribed dimension. We study dimensions of restricted secant of Grassmannians and relate them to the analogous question for secants of Grassmannians via an incidence variety construction. We define a notion of expected dimension and give a formula for the dimension of all restricted secant varieties of Grassmannians that holds if the BDdG conjecture [Baur et al. in Exp Math 16(2):239–250, 2007, Conjecture 4.1] on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay 2, and point out ways to make these calculations more efficient. We also show a potential application to coding theory. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00399-4
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Abstract: Abstract Due to \(\mathbb R_+=(0,\,\infty )\) not being a group under addition, \(L^2(\mathbb R_+)\) admits no traditional wavelet or Gabor frames. This paper addresses a class of modulation-dilation frames ( \(\mathcal MD\) -frames) for \(L^2(\mathbb R_+)\) . We obtain a \(\Theta \) -transform matrix-based expression of adding generators to generate \(\mathcal MD\) -tight frames from a \(\mathcal{M}\mathcal{D}\) -Bessel sequences in \(L^2({\mathbb R}_+)\) ; and present criteria on \(\Phi \) with \(\mathcal{M}\mathcal{D}(\Psi \cup \Phi ,\,a,\,b)\) being a Parseval frame (an orthonormal basis) for an arbitrary Parseval frame sequence (an orthonormal sequence) \(\mathcal{M}\mathcal{D}(\Psi ,\,a,\,b)\) in \(L^2(\mathbb R_+)\) . Some examples are also presented. PubDate: 2024-05-01 DOI: 10.1007/s13348-022-00389-y
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Abstract: Abstract We characterise those Banach spaces X which satisfy that L(Y, X) is octahedral for every non-zero Banach space Y. They are those satisfying that, for every finite dimensional subspace Z, \(\ell _\infty \) can be finitely-representable in a part of X kind of \(\ell _1\) -orthogonal to Z. We also prove that L(Y, X) is octahedral for every Y if, and only if, \(L(\ell _p^n,X)\) is octahedral for every \(n\in {\mathbb {N}}\) and \(1<p<\infty \) . Finally, we find examples of Banach spaces satisfying the above conditions like \({\textrm{Lip}}_0(M)\) spaces with octahedral norms or \(L_1\) -preduals with the Daugavet property. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00394-9
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Abstract: Abstract The set of smooth cubic hypersurfaces in \({{\mathbb {P}}}^n\) is an open subset of a projective space. A compactification of the latter which allows to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points is termed a 1–complete variety of cubic hypersurfaces, in analogy with the space of complete quadrics. Imitating the work of Aluffi for plane cubic curves, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. In the end, we derive the desired numbers in the case of cubic surfaces. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00401-z
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Abstract: Abstract The depth of squarefree powers of a squarefree monomial ideal is introduced. Let I be a squarefree monomial ideal of the polynomial ring \(S=K[x_1,\ldots ,x_n]\) . The k-th squarefree power \(I^{[k]}\) of I is the ideal of S generated by those squarefree monomials \(u_1\cdots u_k\) with each \(u_i\in G(I)\) , where G(I) is the unique minimal system of monomial generators of I. Let \(d_k\) denote the minimum degree of monomials belonging to \(G(I^{[k]})\) . One has \({\text {depth}}(S/I^{[k]}) \ge d_k -1\) . Setting \(g_I(k) = {\text {depth}}(S/I^{[k]}) - (d_k - 1)\) , one calls \(g_I(k)\) the normalized depth function of I. The computational experience strongly invites us to propose the conjecture that the normalized depth function is nonincreasing. In the present paper, especially the normalized depth function of the edge ideal of a finite simple graph is deeply studied. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00392-x
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Abstract: Abstract Let \(f:X\rightarrow Y\) be a semistable fibration between smooth complex varieties of dimension n and m. This paper contains an analysis of the local systems of de Rham closed relative one forms and top forms on the fibers. In particular the latter recovers the local system of the second Fujita decomposition of \(f_*\omega _{X/Y}\) over higher dimensional base. The so called theory of Massey products allows, under natural Castelnuovo-type hypothesis, to study the finiteness of the associated monodromy representations. Motivated by this result, we also make precise the close relation between Massey products and Castelnuovo-de Franchis type theorems. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00397-6
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Abstract: Abstract We construct examples of twice differentiable functions in \({\mathbb {R}}^n\) with continuous Laplacian and unbounded Hessian. The same construction is also applicable to higher order differentiability. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00395-8
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Abstract: Abstract In this paper, we generalize some halfspace type theorems for self-shrinkers of codimension 1 to the case of arbitrary codimension. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00393-w
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Abstract: Abstract In this paper we obtain new estimates for bilinear pseudodifferential operators with symbol in the class \(BS_{1,1}^m\) , when both arguments belong to Triebel-Lizorkin spaces of the type \(F_{p,q}^{n/p}({\mathbb {R}}^n)\) . The inequalities are obtained as a consequence of a refinement of the classical Sobolev embedding \(F^{n/p}_{p,q}({\mathbb {R}}^n)\hookrightarrow \textrm{bmo}({\mathbb {R}}^n)\) , where we replace \(\textrm{bmo}({\mathbb {R}}^n)\) by an appropriate subspace which contains \(L^\infty ({\mathbb {R}}^n)\) . As an application, we study the product of functions on \(F_{p,q}^{n/p}({\mathbb {R}}^n)\) when \(1<p<\infty \) , where those spaces fail to be multiplicative algebras. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00400-0
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Abstract: Abstract Assume that G is a graph with edge ideal I(G). We provide sharp lower bounds for the depth of \(I(G)^2\) in terms of the star packing number of G. PubDate: 2024-05-01 DOI: 10.1007/s13348-023-00398-5
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Abstract: Abstract 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. PubDate: 2024-04-30 DOI: 10.1007/s13348-024-00442-y
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