Stability of syzygy bundles

We show that given integers $N$, $d$ and $n$ such that ${N\ge2}$, ${(N,d,n)\ne(2,2,5)}$, and ${N+1\le n\le\tbinom{d+N}{N}}$, there is a family of $n$ monomials in $K[X_0,\ldots,X_N]$ of degree $d$ such that their syzygy bundle is stable. Case ${N\ge3}$ was obtained independently by Coand\v{a} with a different choice of families of monomials [Coa09]. For ${(N,d,n)=(2,2,5)}$, there are $5$ monomials of degree~$2$ in $K[X_0,X_1,X_2]$ such that their syzygy bundle is semistable.Comment: 16 pages, to appear in the Proceedings of the American Mathematical Societ

Stability and Unobstructedness of Syzygy Bundles

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,..., d_n}$ on \PP^N defined as the kernel of a general epimorphism \xymatrix{\phi:\cO(-d_1)\oplus...\oplus\cO(-d_n)\ar[r] &\cO} is (semi)stable. In this note, we restrict our attention to the case of syzygy bundles $E_{d,n}$ on \PP^N associated to $n$ generic forms $f_1,...,f_n\in K[X_0,X_1,..., X_N]$ of the same degree $d$. Our first goal is to prove that $E_{d,n}$ is stable if $N+1\le n\le\tbinom{d+2}{2}+N-2$. This bound improves, in general, the bound $n\le d(N+1)$ given by G. Hein in \cite{B}, Appendix A. In the last part of the paper, we study moduli spaces of stable rank $n-1$ vector bundles on \PP^N containing syzygy bundles. We prove that if $N+1\le n\le\tbinom{d+2}{2}+N-2$ and $N\ne 3$, then the syzygy bundle $E_{d,n}$ is unobstructed and it belongs to a generically smooth irreducible component of dimension $n\tbinom{d+N}{N}-n^2$, if $N \geq 4$, and $n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2$, if N=2.Comment: 32 pages, minor change

Juntas / Juntes

Multitud d'autores i autors aporten píndoles de microliteratura, de gèneres diversos, al tercer volum amb què l'Institut Universitari d'Estudis Feministes i de Gènere Purificación Escribano vol respondre al desafiament per l'eradicació de la violència contra les dones.Tercer desafío por la erradicación de la violencia contra las mujeres del Institut Universitari d'Estudis Feministes i de Gènere Purificación Escribano de la Universitat Jaume I a través de microliteratura

Microrelatario

Multitud d'autores i autors aporten píndoles de microliteratura, de gèneres diversos, al segon volum amb què l'Institut Universitari d'Estudis Feministes i de Gènere Purificación Escribano vol respondre al desafiament per l'eradicació de la violència contra les dones