We introduce the notion of rooftop flip, namely a small modification among normal projective varieties which is modeled by a smooth projective variety of Picard number 2 admitting two projective bundle structures. Examples include the Atiyah flop and the Mukai flop, modeled respectively by P1×P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}^1\ imes \\mathbb {P}^1$$\\end{document} and by PTP2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}\\left( T_{\\mathbb {P}^2}\\right) $$\\end{document}. Using the Morelli-Włodarczyk cobordism, we prove that any smooth projective variety of Picard number 1, endowed with a C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb C}^*$$\\end{document}-action with only two fixed point components, induces a rooftop flip.

We describe a combinatorial condition on a graphwhich guarantees that all powers of its vertex cover ideal are componentwise linear. Then motivated by Eagon and Reiner’s Theorem we study whether all powers of the vertex cover ideal of a Cohen-Macaulay graph have linear free resolutions. After giving a complete characterization of Cohen-Macaulay cactus graphs (i.e., connected graphs in which each edge belongs to at most one cycle) we show that all powers of their vertex cover ideals have linear resolutions.

In this paper, we are mainly concerned with the blow-up algebras of the secant varieties of balanced rational normal scrolls. In the first part, we give implicit defining equations of their associated Rees algebras and fiber cones. Consequently, we can tell that the fiber cones are Cohen–Macaulay normal domains. Meanwhile, these fiber cones have rational singularities in characteristic zero, and are F-rational in positive characteristic. The Gorensteinness of the fiber cones can also be characterized. In the second part, we compute the Castelnuovo–Mumford regularities and $${\varvec{a}}$$ -invariants of the fiber cones. We also present the reduction numbers of the ideals defined by the secant varieties.

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator overline{partial } on spaces {mathcal {E}}{mathcal {V}}(varOmega ,E) of {mathcal {C}}^{infty }-smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights {mathcal {V}}. Vector-valued means that these functions have values in a locally convex Hausdorff space E over {mathbb {C}}. We derive a counterpart of the Grothendieck-Köthe-Silva duality {mathcal {O}}({mathbb {C}}setminus K)/{mathcal {O}}({mathbb {C}})cong {mathscr {A}}(K) with non-empty compact Ksubset {mathbb {R}} for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of overline{partial }:{mathcal {E}} {mathcal {V}}(varOmega ,E)rightarrow {mathcal {E}}{mathcal {V}} (varOmega ,E) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on {mathcal {E}}{mathcal {V}}(varOmega ,{mathbb {C}}).

Any $${\mathcal {S}} \in \mathfrak {sp}(1,{\mathbb {R}})$$ induces canonically a derivation S of the Heisenberg Lie algebra $${\mathfrak {h}}$$ and so, a semi-direct extension $$G_{{\mathcal {S}}}=H \rtimes \exp ({\mathbb {R}}S)$$ of the Heisenberg Lie group H (Muller and Ricci in Invent Math 101: 545–582, 1990). We shall explicitly describe the connected, simply connected Lie group $$G_{{\mathcal {S}}}$$ and a family $$g_a$$ of left-invariant (Lorentzian and Riemannian) metrics on $$G_{{\mathcal {S}}}$$ , which generalize the case of the oscillator group. Both the Lie algebra and the analytic description will be used to investigate the geometry of $$(G_{{\mathcal {S}}},g_a)$$ , with particular regard to the study of nontrivial Ricci solitons.

In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category $\mathcal{C}$ and for arbitrary groups $H\le G_1\times G_2$, is there an object $\phi \colon A_1 \rightarrow A_2$ in $\operatorname{Arr}(\mathcal{C})$ such that $\operatorname{Aut}_{\operatorname{Arr}(\mathcal{C})}(\phi) = H$, $\operatorname{Aut}_{\mathcal{C}}(A_1) = G_1$ and $\operatorname{Aut}_{\mathcal{C}}(A_2) = G_2$? We are interested in solving this problem when $\mathcal C =\mathcal{H}oTop_*$, the homotopy category of pointed topological spaces. To that purpose, we first settle that question in the positive when $\mathcal C = \mathcal{G}raphs$. Then, we construct an almost fully faithful functor from $\mathcal{G}raphs$ to $\operatorname{CDGA}$, the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when $\mathcal C = \operatorname{CDGA}$ and, as long as we work with finite groups, when $\mathcal C =\mathcal{H}oTop_*$. Some results on representability of concrete categories are also obtained.