Hecke curves in Frobenius strata of moduli space of rank 2 vector bundles
Lingguang Li, Hongyi Zhang
TL;DR
This work studies the Frobenius stratification of the moduli space $\mathcal{M}^s_X(2,\mathcal{L})$ of rank-2 stable bundles on a genus $g\ge2$ curve $X$ over an algebraically closed field of characteristic $2$, by tracking stability under Frobenius pullback via Harder–Narasimhan polygons. It defines Frobenius strata $\mathcal{M}_j$ and their determinant-labeled counterparts $\mathcal{M}_j(\mathcal{L})$, drawing on results from $\mathrm{JRXY06}$ and prior work to relate submodules of $F_*L$ and extensions $0 \to E \to F_*L \to \mathbf{k}(x) \to 0$ to destabilization phenomena. The paper then leverages Hecke transformations to construct rational curves (Hecke curves) within the moduli, establishing that in characteristic $2$ there are no rational curves in the top stratum $\mathcal{M}_{g-1}$; every rational curve in $\mathcal{M}^s_X(2,d)$ lies in some fixed determinant stratum $\mathcal{M}^s_X(2,\mathcal{L})$, and through any point in the lower stratum $\mathcal{M}_{g-2}(\mathcal{L}) \setminus \mathcal{M}_{g-1}(\mathcal{L})$ there exists a Hecke curve contained in that locus. This connects Frobenius instability to explicit geometric curves in the moduli space, extending the minimal-rational-curve picture to positive characteristic via explicit Hecke-curve constructions.
Abstract
Let $k$ be an algebraically closed field with characteristic $2$, and let $X$ be a smooth projective algebraic curve of genus $g \geqslant 2$ over $k$. Let $\mathcal{M}^s_X(2,\mathcal{L})$ be the moduli space of rank $2$ stable vector bundles with determinant $\mathcal{L}$ on $X$. The Frobenius stratification measures the instability of bundles in $\mathcal{M}^s_X(r,\mathcal{L})$ under pullback by the Frobenius map. We show that there exists a Frobenius stratum in $\mathcal{M}^s_X(2,\mathcal{L})$ which is covered by Hecke curves.