On threefolds with \(K^3=2p_q-6\) (Q1884363)

It is a classical result that the invariants of a minimal surface of general type satisfy \(K^2\geq 2p_g-4\) and that surfaces satisfying \(K^2=2p_g-4\) are mapped by the canonical map \(2-\)to\(-1\) onto a surface of minimal degree in \(\mathbb{P}^{p_g-1}\). Analogously, for \(n\geq 3\) if \(X\) is a normal \(n-\)dimensional variety of general type with \(K_X\) nef, then \(K_X^n\geq 2(p_g(X)-n)\), provided that the canonical map of \(X\) is generically finite. Moreover, if equality holds, then the canonical map of \(X\) is \(2-\)to\(-1\) onto a variety of minimal degree in \(\mathbb{P}^{p_g-1}\). The author studies the above situation in the following cases: a) \(K_X^n=2(p_g(X)-n)=2,4\), \(n\geq 3\); b) \(n=3\) and \(K^3_X=2p_g(X)-6\). In case a) she proves that \(|K_X|\) is free and the canonical map of \(X\) is a double cover of \(\mathbb{P}^n\), when \(K_X^n=2\), or of a quadric, when \(K_X^n=4\). In case b), the canonical map of \(X\) is of degree 2 onto a variety \(W\) of minimal degree in \(\mathbb{P}^{p_g-1}\). When \(W\) is the cone over the Veronese surface, there exists a \(2-\)dimensional family of examples (\(p_g(X)=7\), in this case). In general, for every value of \(k\geq 3\) the author constructs a family of examples with \(p_g(X)=k+3\) and \(W\) a rational scroll.