Groupoid Quantales: a non \'etale setting

It is well known that if G is an \'etale topological groupoid then its topology can be recovered as the sup-lattice generated by G-sets, i.e. by the images of local bisections. This topology has a natural structure of unital involutive quantale. We present the analogous construction for any non \'etale groupoid with sober unit space G_0. We associate a canonical unital involutive quantale with any inverse semigroup of G-sets which is also a sheaf over G_0. We introduce axiomatically the class of quantales so obtained, and revert the construction mentioned above by proving a representability theorem for this class of quantales, under a natural spatiality condition

On the maximum rank of a real binary form

We show that a real homogeneous polynomial f(x,y) with distinct roots and degree d greater or equal than 3 has d real roots if and only if for any (a,b) not equal to (0,0) the polynomial af_x+bf_y has d-1 real roots. This answers to a question posed by P. Comon and G. Ottaviani, and shows that the interior part of the locus of degree d binary real binary forms of rank equal to d is given exactly by the forms with d real roots.Comment: To appear in Annali di Matematica Pura ed Applicat

PGL(2) actions on Grassmannians and projective construction of rational curves with given restricted tangent bundle

We give an explicit parametrization of the Hilbert schemes of rational curves C in P^n having a given splitting type of the restricted tangent bundle from P^n to C. The adopted technique uses the description of such curves as projections of a rational normal curve from a suitable linear vertex and a classification of those vertices that correspond to the required splitting type of the restricted tangent bundle. This classification involves the study of a suitable PGL(2) action on the relevant Grassmannian variety

A Note on Super Koszul Complex and the Berezinian

We construct the super Koszul complex of a free supercommutative $A$-module $V$ of rank $p|q$ and prove that its homology is concentrated in a single degree and it yields an exact resolution of $A$. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to $\Pi^{p+q} A$, with $\Pi$ the parity changing functor. Finally, we show that, given an automorphism of $V$, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of $V$.Comment: 13 pages, reference adde

One-Dimensional Super Calabi-Yau Manifolds and their Mirrors

We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY's having reduced manifold equal to $\mathbb{P}^1$, namely the projective super space $\mathbb{P}^{1|2}$ and the weighted projective super space $\mathbb{WP}^{1|1}_{(2)}$. Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces $\mathbb P^{n|m}$. We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of $\mathbb{P}^{1|2}$, whose automorphism group turns out to be larger than the projective general linear supergroup. By considering the cohomology of the super tangent sheaf, we compute the deformations of $\mathbb{P}^{1|m}$, discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that $\mathbb{P}^{1|2}$ is self-mirror, whereas $\mathbb{WP} ^{1|1}_{(2)}$ has a zero dimensional mirror. Also, the mirror map for $\mathbb{P}^{1|2}$ naturally endows it with a structure of $N=2$ super Riemann surface.Comment: 50 pages. Accepted for publication in JHE