Bitangents to a quartic surface and infinitesimal Torelli (Q6561315)

The study of curves using their associated Jacobian varieties is a classical problem in algebraic geometry, which was introduced and studied by R. Torelli in the first decade of the XX century. Afterwords, in the '60s, \textit{P. A. Griffiths} [Ann. Math. (2) 90, 460--495, 496--541 (1969; Zbl 0215.08103)], generalized the problem considering period maps. He investigated whether for a simply connected surface of general type the period map is an embedding, i.e., if the infinitesimal Torelli holds. The subsequent literature on infinitesimal Torelli is vast. In particular, \N\textit{H. Flenner} [Math. Z. 193, 307--322 (1986; Zbl 0613.14010)] showed that infinitesimal Torelli holds for any smooth complete intersection in \(\mathbb{P}^n\) except of hypersurfaces of degree three in \(\mathbb{P}^3\) and intersections of two quadrics of even dimension. \N\textit{K. Konno} [Tôhoku Math. J. (2) 38, No. 1--2, 609--624 (1986; Zbl 0627.14009); Tôhoku Math. J. (2) 42, No. 3, 333--338 (1990; Zbl 0727.14003); Tôhoku Math. J. (2) 43, No. 4, 557--568 (1991; Zbl 0824.14006)] proved the inifinitesimal Torelli property in the case of certain complete intersections inside homogeneous varieties. More recently \N\textit{E. Fatighenti} et al. [J. Geom. Phys. 139, 1--16 (2019; Zbl 1441.14037)] solved the infinitesimal Torelli problem for 3-dimensional quasi-smooth \(\mathbb{Q}\)-Fano hypersurfaces with at worst terminal singularities. Thus the study of the infinitesimal Torelli problem is an active area of the algebraic geometry, which gives a lot of information about the Hodge structures and monodromy. This article deals with the infinitesimal Torelli problem for the surface of the bitangents to a quartic surface, and gives examples of surfaces of general type with standard geometrical assumptions, where the infinitesimal Torelli does not hold. Moreover, the authors analyzed also its canonical map and the dimension of its deformation space, giving a complete picture about the geometry of surface of the bitangents to a quartic surface.\N\NIn the first part of this work, the authors built \(S\subset \mathrm{Gr}(2,4)\), the variety of bitangent lines to the generic quadric \(X\subset\mathbb{P}^3\). To do so they first studied the variety of lines \(S_X\) in \(Q\) the quartic double solid, which is the Fano threefold 1--12 in the database \textit{Fanography}, by P. Belmans. In this way \(S\) can be obtained as the image of the forgetful map between schemes \(f:S_X\to S\). This description with the restriction to \(S\) of the universal sequence:\N\[\N0\to\mathcal{S}^{\vee}\to V_4\otimes\mathcal{O}_{\mathrm{Gr}(2,4)}\to\mathcal{Q}\to 0\N\]\Ninduces a natural diagram of morphisms described in Subsection 3.4.1. The study of the map involved in the above diagram gives an explicit way to compute the class of \(S\) in the Chow ring of \(\mathrm{Gr}(2,4)\) (Lemma 3.4.7), the invariants (Theorem 3.4.8), and to show that if \(X\) is a smooth quartic containing no line, then there are no rational lines on \(S\) (Theorem 3.6.1).\N\NThis construction of \(S\) is also the core of the next results shown in the paper. In fact, the tangent bundle sequence gives:\N\[\N0\to\mathcal{O}_S\to \mathrm{Sym}^2(\mathcal{Q}_S)\otimes\mathcal{O}_S(\sigma)\to\Omega^1_S\to 0\N\]\Nwhere \(\sigma\in Pic^2(S)\). And, looking to the homomorphisms induced by the extension class \(\zeta\in H^1(S,T_S)\) of the above sequence, they could prove that \(S\subset \mathrm{Gr}(2,4)\) is a counterexample to infinitesimal Torelli.\N\NSection 5 and 6 are devoted to the canonical map \(\phi_{|K_S|}\) and its image of the surfaces of bitangents. In particular, it is shown that:\N\[\N\phi_{|K_S|}:S\to\mathbb{P}(H^0(S,\mathcal{O}_S(K_S))^{\vee})\N\]\Nis an embedding and that the image is not 2-normal.\N\NIn the end, the final section, is focused on computing \(h^1(S,\Omega^1_S\otimes_{\mathcal{O}_S}\omega_S)=h^1(S,T_S)\). The strategy to do so is tensoring the tangent bundle sequence by \(\omega_S\) and noting that computing \(h^1(S,\Omega^1_S\otimes_{\mathcal{O}_S}\omega_S)\) is equivalent to compute \(h^0(S,\mathrm{Sym}^2(\mathcal{Q_S})\otimes\mathcal{O}_S(\sigma)\otimes_{\mathcal{O}_S}\omega_S)\), which can be computed using the diagram in 3.4.1.\N\NIn particular, all the results in the paper can be summarized as:\N\NLet \(X\) be a quartic surface which contains no lines. Then the surface \(S\) which parametrizes its bitangent lines is a smooth surface of general type with very ample canonical sheaf. It contains no rational curves, and the infinitesimal Torelli property does not hold for it. Moreover the canonical model is not 2-normal and the dimension of \(H^1(S,T_S)\) is 20.