Spectral Endomorphism Factorization and Exceptional Degree-Two Root Correspondences in Codimension-Three Lefschetz Failure

Abstract

Let $X$ be a smooth rational surface obtained by resolving the generically reduced quadratic-root correspondence attached to a controlled codimension-three weak Lefschetz failure. The preceding Pfaffian analysis produces a pulled-back rank-two quotient bundle $Q$, a nowhere-zero section $q\in H^0(X,\Sym^2Q\otimes\OO_X(M))$, a finite flat root double cover $p:\Rcal\to X$, and a rank-two quotient $0\longrightarrow\OO_X(-M)\longrightarrow\Sym^2Q\longrightarrow G\longrightarrow0$ with $4c_2(G)-c_1(G)^2=3$. We study the first remaining correspondence degree $e=H\cdot M=2$. Writing $\Lambda=\OO_{\Rcal}(2\xi)$, we prove the spectral realization $G\cong p_*\Lambda$ and an exact sequence $0\longrightarrow p_*\OO_{\Rcal}\longrightarrow\End G$ $\longrightarrow p_*\Theta\longrightarrow0, $$\qquad \Theta=\Lambda^{\otimes2}\otimes p^*(\det G)^{-1}.$ Every non-scalar endomorphism is then represented by an explicit spectral factorization. A nilpotent endomorphism yields a square factorization indexed by a divisor class $T$ and forces the half-branch class to be effective. A semisimple endomorphism splits $G$ and yields two orthogonal quotient binary quadratics whose dual discriminants factor the branch section. For $e=2$, the nef kernel class is orthogonal to the half-branch class. On an integral root cover this excludes the nilpotent branch. In the split branch, the quotient of smaller plane degree descends to a fixed ternary symmetric tensor. Rank three contradicts the exceptional twisting, rank two forces a double-line component of the branch divisor, and rank one has identically zero discriminant. Consequently, if the connected root double cover is normal, then $G$ is simple. Riemann--Roch gives $\chi(\End G)=1$, while a degree-window argument gives $\Ext^2(G,G)=0$ for $e=2$; hence $G$ is exceptional. Normality is retained as a hypothesis. We do not classify nonnormal degree-two correspondences, treat degrees $e\ge3$, exclude the entire quadratic-apolar stratum, or prove the weak Lefschetz property in full codimension three.