Prym-Tjurin constructions on cubic hypersurfaces
The author gives a Prym-Tjurin construction for a cubic hypersurface. In general, for an abelian group \(\Lambda\) and an endomorphism \(\sigma\) of \(\Lambda\), which satisfies the quadratic relation \((\sigma-1)(\sigma+q-1)=0\), the Prym-Tjurin part \(P(\Lambda,\sigma)\) of \(\Lambda\) is defined to be \(\mathrm{Im}(\sigma-1)\). Let \(X\subset\mathbb{P}_{\mathbb{C}}^{n+1}\) be a smooth cubic hypersurface of dimension \(n\geq 3\). Let \(C\subset X\) be a general rational curve of degree \(e\geq 2\), and let \(S_C\) be a natural resolution of the space of lines on \(X\) that meet \(C\). Let \(p_C:P_C\rightarrow S_C\) be the family of lines parametrized by \(S_C\) and let \(q_C:P_C\rightarrow X\) be the natural morphism. Then one has the cylinder homomorphism \(\Phi_C=(p_C)_{*}(q_C)^{*}:H^n(X, \mathbb{Z})\rightarrow H^{n-2}(S_C,\mathbb{Z})\). The author defines an incidence correspondence which induces an action \(\sigma\) on the cohomology groups and the Chow groups of \( S_C\) and proves the following main result of this paper: The action \(\sigma\) satisfies the quadratic relation \((\sigma-1)(\sigma+2e-1)=0\), and the map \(\Phi_C\) induces an isomorphism of Hodge structures \(\Phi_C:H^n(X,\mathbb{Z})_{\mathrm{prim}}\rightarrow P(H^{n-2}(S_C,\mathbb{Z})^{\circ},\sigma)(-1)\), where \(H^{n-2}(S_C,\mathbb{Z})^{\circ}\) denotes a sub-Hodge structure of the cohomology group and ``\((-1)\)'' means shifting of degree by \((1,1)\). A similar isomorphism is shown to hold between the Chow group of \(X\) and that of \(S_C\).
scientific article; zbMATH DE number 6359323