U. N. Bhosle | Searcharxiv

arXiv2014

On linear systems and a conjecture of D. C. Butler

Let $C$ be a smooth irreducible projective curve of genus $g$ and $L$ a line bundle of degree $d$ generated by a linear subspace $V$ of $H^0(L)$ of dimension $n+1$. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map $V\otimes{\mathcal O}_C\to L$ and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

math.AG↗

arXiv2007

On coherent systems of type (n,d,n+1) on Petri curves

We study coherent systems of type $(n,d,n+1)$ on a Petri curve $X$ of genus $g\ge2$. We describe the geometry of the moduli space of such coherent systems for large values of the parameter $α$. We determine the top critical value of $α$ and show that the corresponding ``flip'' has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of $α$, proving in many cases that the condition for non-emptiness is the same as for large $α$. We give some detailed results for $g\le5$ and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known.

math.AG↗