Complete intersections of quadrics and complete intersections on Segre varieties with common specializations
An exercise of \textit{A. Beauville} [Astérisque 54, 1--172 (1978; Zbl 0394.14014)] asks the reader to show that the image of a smooth surface \(S \subset \mathbb P^1 \times \mathbb P^2\) of bidegree \((2,3)\) in \(\mathbb P^5\) under the Segre embedding is a limit of complete intersections of three quadric hypersurfaces. Along with carrying out this exercise, the authors ask more generally when images of complete intersection surfaces in the Segre embedded product \(\mathbb P^1 \times \mathbb P^k \hookrightarrow \mathbb P^{2k+1}\) can belong to the same Hilbert scheme as surfaces which are complete intersections of quadrics. Their first theorem shows how rare this is: other than the case \(k=2\) above, they prove that if \(T\) is a smooth surface with ample canonical system embedded as a complete intersection in \(\mathbb P^1 \times \mathbb P^k\) whose image under the Segre embedding in \(\mathbb P^{2k+1}\) is in the same Hilbert scheme as a complete intersection \(S\) of quadrics, then \(k=3, c_1^2 (S) = c_1^2 (T) = 2^7\), \(c_2 (S) = c_2 (T) = 2^8\) and \(T\) is a complete intersection of type \((0,4), (4,2)\) on \(\mathbb P^1 \times \mathbb P^3\). The authors then examine the moduli spaces of these families. Let \(\mathfrak M\) be the moduli space of simply connected minimal surfaces of general type with \(c_1^2=2^7\) and \(c_2=2^8\). Then the Kuranishi space of smooth complete intersections \(S\) of five quadrics in \(\mathbb P^7\) is smooth of dimension \(92\), as is the corresponding component \(\mathfrak M_S \subset \mathfrak M\). The restricted Kuranishi space \(\mathfrak M^\prime_T\) for smooth complete intersections \(T\) of type \((0,4), (4,2)\) in \(\mathbb P^1 \times \mathbb P^3\) is smooth of dimension \(95\) and the Kuranishi space \(\mathfrak M_T\) is smooth. Finally, they show that the closure of \(\mathfrak M_T\) in \(\mathfrak M\) does not meet the component \(\mathfrak M_S\): no smooth type \(T\) surface degenerates to an intersection of five quadrics in \(\mathbb P^7\). The authors note that in the family of surfaces \(T\) described, only \(65\) of the \(95\) dimensions are accounted for by the obvious deformations of the surface \(T\) (the deformation of complete intersections in \(\mathbb P^1 \times \mathbb P^3\)), an instance of the Murphy's law philosophy of \textit{R. Vakil} [Invent. Math. 164, 569--590 (2006; Zbl 1095.14006)]. In an appendix they explain the additional \(30\) parameters in terms of deformations of the normal bundle of \(T\) in \(\mathbb P^1 \times \mathbb P^3\).
scientific article; zbMATH DE number 7336979
scientific article