A construction of wobbly divisors of nilpotency index greater than 2
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For a smooth projective curve $C$, let $U_C(n, d)$ be the moduli space of semistable bundles over $C$ of rank $n$ and degree $d$. A vector bundle $V$ in $U_C(n,d)$ is called \textit{wobbly} if there is a nonzero nilpotent $K_C$-valued endomorphism $\phi: V \to V \otimes K_C$. Drinfeld's conjecture predicts that the wobbly locus has pure codimension one. This has been proven for $n=2$, for which case the involved endomorphisms have nilpotency index 2. In this paper, we construct irreducible wobbly components of codimension 1 and nilpotency index greater than 2, under a certain numerical condition on $n$ and $d$.
Abstract
For a smooth projective curve $C$, let $U_C(n, d)$ be the moduli space of semistable bundles over $C$ of rank $n$ and degree $d$. A vector bundle $V$ in $U_C(n,d)$ is called \textit{wobbly} if there is a nonzero nilpotent $K_C$-valued endomorphism $\phi: V \to V \otimes K_C$. Drinfeld's conjecture predicts that the wobbly locus has pure codimension one. This has been proven for $n=2$, for which case the involved endomorphisms have nilpotency index 2. In this paper, we construct irreducible wobbly components of codimension 1 and nilpotency index greater than 2, under a certain numerical condition on $n$ and $d$.