Hecke curves in Frobenius strata of moduli space of rank 2 vector bundles

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.

💡 Research Summary

The paper studies the geometry of Frobenius strata inside the moduli space of rank‑2 stable vector bundles with fixed determinant on a smooth projective curve X of genus g ≥ 2 over an algebraically closed field k of characteristic 2. For a bundle E the absolute Frobenius morphism F: X→X induces a pull‑back F⁎E whose Harder–Narasimhan polygon (HNP) measures the loss of stability under Frobenius. Fixing the determinant L∈Pic⁽ᵈ⁾(X), the authors consider the Frobenius stratification \