Collectanea Mathematica

The aim of this paper is twofold. We first strongly improve our previous main result Choi et al. (Proc Am Math Soc 146(8):3233–3248, 2018, Theorem 3.1), concerning classification of irreducible components of the Brill–Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees d, with at least $$d-2g+4$$ independent global sections, on a general $$\nu $$ -gonal curve C of genus g. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which turn out to be special and stable.

We focus on the rational cohomology of Cornalba’s moduli space of spin curves of genus 1 withn marked points. In particular, we show that both its first and its third cohomology group vanish and the second cohomology group is generated by boundary classes.

Let R be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do not assume that their quotient has finite length. In this paper, we develop various sufficient numerical criteria for when the tight closures of these ideals (or submodules) match. For some of the criteria we only prove sufficiency, while some are shown to be equivalent to the tight closures matching. We compare the various numerical measures (in some cases demonstrating that the different measures give truly different numerical results) and explore special cases where equivalence with matching tight closure can be shown. All of our measures derive ultimately from Hilbert–Kunz multiplicity.

A categorical formalism for directed graphs is introduced, featuring natural notions of morphisms and subgraphs, and leading to two elementary descriptions of the free-properad monad, first in terms of presheaves on elementary graphs, second in terms of groupoid-enriched hypergraphs.

We introduce a class of pseudodifferential operators in the anisotropic setting induced by an expansive dilation A which generalizes the classical isotropic class $${S^{m}_{\gamma, \delta}}$$ of inhomogeneous symbols. We extend a well-known L 2-boundedness result to the anisotropic class $${S_{\delta, \delta}^0(A)}$$ , 0 ≤ δ < 1. As a consequence, we deduce that operators with symbols in the anisotropic class $${S^0_{1,0}(A)}$$ are bounded on L p spaces, 1 < p < ∞.

The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.

For a sequence of complex numbers $${\Lambda}$$ we consider the restriction operator $${R_{\Lambda}}$$ defined on Paley–Wiener spaces $${PW_{\tau}^{p}}$$ (1 < p < ∞). Lyubarskii and Seip gave necessary and sufficient conditions on $${\Lambda}$$ for $${R_{\Lambda}}$$ to be an isomorphism between $${PW_{\tau}^{p}}$$ and a certain weighted l p space. The Carleson condition appears to be necessary. We extend their result to N–Carleson sequences (finite unions of N disjoint Carleson sequences). More precisely, we give necessary and sufficient conditions for $${R_{\Lambda}}$$ to be an isomorphism between $${PW_{\tau}^{p}}$$ and an appropriate sequence space involving divided differences.