Noether-Lefschetz, space curves and mathematical instantons
The author formulates, interrelates and in special cases proves conjectures pertaining to the three topics of the title. Let \(G_{(n,s)}\) denote the maximum genus of a smooth curve of degree \(n\) in \(\mathbb{P}^ 3\) (complex projective 3-space) not contained in a surface of degree less than \(s\), and let \(H_{n,G_{(n,s)}}\) denote the Hilbert scheme of smooth curves of degree \(n\) and genus \(G_{(n,s)}\) in \(\mathbb{P}^ 3\). Conjecture (H). If \({(s^ 2 + 4s + 6) \over 6} \leq n < {(s^ 2 + 4s + 6) \over 3}\) then \(H_{n,G_{(n,s)}}\) is pure dimensional of dimension \(4n\). This conjecture has been proven for certain special cases, e.g. \(s=2\). Next recall that an instanton bundle on \(\mathbb{P}^ 3\) is a rank 2 bundle \(F\) with \(c_ 1(F) = 0\), \(H^ 0(F) = H^ 1(F(-2)) = 0\). There is a well defined moduli space \(MI(0,c_ 2)\) of instantons with \(c_ 2(F) = c_ 2\). Conjecture \((\text{I}')\). Every component of \(MI (0,c_ 2)\) has dimension \(8c_ 2 -3.\) (Conjecture I is the conjecture that \(MI (0,c_ 2)\) is irreducible, and is known to imply \(\text{I}'\), where the notation. It has been proven for \(c_ 2 \leq 4.)\) A further technical conjecture, attributed to \textit{R. Hartshorne} [in: Vector bundles and differential equations, Proc., Nice 1979, Prog. Math. 7, 83-112 (1980; Zbl 0452.14005)] is: Conjecture (T). Let \(W\) be a component of \(M(0, c_ 2)\) and let \(F\) be a general element of \(W\). Then \(H^ 0 (F(t)) = 0\) for \(t \leq \sqrt {3c_ 2 + 1} - 2\). Let \(S(d)\) be the space of smooth surfaces of degree \(d\) in \(\mathbb{P}^ 3\) and let \(NL(d)\) be the Noether-Lefschetz locus, i.e., the subset of \(S(d)\) consisting of smooth surfaces whose Picard group is larger than \(\mathbb{Z}\). It is known that \(NL(d)\) is the union of countably many closed proper subsets of \(S(d)\), called components. A component of \(NL(d)\) is general if its codimension in \(S(d)\) is the expected value \(\left( \begin{smallmatrix} d - 1 \\ 3 \end{smallmatrix} \right)\), otherwise special. Green and Ciliberto conjectured that for a generic element \(S\) of a special component of \(NL(d)\), some canonical divisor on \(S\) has a component whose class is not a multiple of the hyperplane class. This would imply that there are only finitely many special components of \(NL(d)\), but this is now known to be false [\textit{C. Voisin}, C. R. Acad. Sci., Paris, Sér. I 313, 685-687 (1991; Zbl 0751.14020)]. Nevertheless, there is some hope that restricted versions of the Green-Ciliberto conjecture may be true. In the present paper the author introduces the notion of regular component \(W(d)\) of \(NL(d)\); the definition involves the choice of a component \(W\) of the Hilbert scheme of smooth curves of \(\mathbb{P}^ 3\). If \(C\) is a curve representing the generic point of \(W\), and \(S\) a general surface of degree \(d\) containing \(C\) then \(C^ 2\) means the self-intersection of \(C\) in \(S\). Conjecture (GCR). The Green-Ciliberto conjecture holds for special regular components of \(NL(d)\) with \(C^ 2 \leq 0\). The principal results of the paper are the following: Theorem. \((d \geq 7)\). Conjecture (H) is equivalent to conjecture (GCR) for components of \(NL(d)\) with \(C^ 2 \leq - 2\). Theorem. \((d \geq 7)\). Conjecture (GCR) is true for components of \(NL(d)\) with \(C^ 2 = 0\) if \(d \neq 10, d\equiv 1\pmod 3\). Let \({\mathcal J}_ C\) be the ideal of \(C\) in \({\mathcal O}_{\mathbb{P}^ 3}\). Theorem. Suppose \(d \equiv 0,2 \pmod 6\), \(d \geq 8\), and let \(c_ 2 = {d^ 2 - 8d + 12 \over 12}\). Then Conjecture \((\text{I}')\) implies conjecture (GCR) for components of \(NL(d)\) with \(H^ 1({\mathcal J}_ C({d \over 2} - 2)) = 0\) and \(C^ 2 = 0\). Conversely, conjecture (GCR) for components of \(NL(d)\) with \(C^ 2 = 0\) and \(H^ 1({\mathcal J}_ C ({d \over 2} - 2)) = 0\), together with conjecture (T), implies conjecture \((\text{I}')\).
scientific article; zbMATH DE number 562487
scientific article