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In the projectivized cotangent space at a general point $E$ of ${\\cal SU}^s_C(r,d)$, there exists a distinguished hypersurface ${\\cal S}_E$ consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at $E$, there exists a distinguished subvariety ${\\cal C}_E$ consisting of vectors tangent to Hecke curves in ${\\cal SU}^s_C(r,d)$ through $E$. 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