Stability and singularities of relative hypersurfaces (Q2797851)

The authors study stability properties and singularities of relative hypersurfaces over complex curves. Let \(\mathcal{E}\) be a vector bundle of rank \(r\geq3\) and degree \(d\) on a smooth projective curve \(B\) of genus \(b\) and consider the relative projective bundle \(\mathbb{P}: = \mathbb{P}_{B}(\mathcal{E})\) with \(\pi : \mathbb{P} \rightarrow B\). Denote by \(\mathcal{O}_{\mathbb{P}}(1)\) the tautological sheaf. Let \(X\) be a relative (possibly singular) hypersurface, i.e. \(X\) is an effective divisor in the linear system \(|\mathcal{O}_{\mathbb{P}}(k) \otimes \pi^{*} \mathcal{M}^{-1}|\) with \(k > 0\) and \(\mathcal{M}\) an invertible sheaf on \(B\) of degree \(y\).NEWLINENEWLINEThe first result of the paper is devoted to the \(f\)-positivity. Given a fibered \(n\)-dimensional variety \(g : Y \rightarrow T\) over a smooth curve \(T\) and given a line bundle \(\mathcal{L}\) on \(Y\), we say that \(\mathcal{L}\) is \(g\)-positive if the following inequality holds: NEWLINE\[NEWLINE\mathcal{L}^{n} \geq n \frac{\mathcal{L}^{n-1}|_{F}}{h^{0}(F, \mathcal{L}|_{F})} \text{deg} \, g_{*}\mathcal{L}.NEWLINE\]NEWLINENEWLINENEWLINEWhen fibers are of general type, then we can consider the \(g\)-positivity of the relative canonical sheaf \(\omega_{g} = \omega_{Y} \otimes \omega_{T}^{-1}\): NEWLINE\[NEWLINE K_{g}^{n} \geq n \frac{K^{n-1}_{F}}{p_{g}(F)} \text{deg} \, g_{*} \omega_{g} \quad \quad (\star)NEWLINE\]NEWLINE This inequality is usually called the slope inequality. Moreover, for a vector bundle \(\mathcal{E}\) we define its slope by \(\mu = \text{deg} \, \mathcal{E} / r\).NEWLINENEWLINEThe first result of the paper can be formulated as follows.NEWLINENEWLINE{ Theorem 1.} Let \(X \in | \mathcal{O}_{\mathbb{P}}(k) \otimes \pi^{*} \mathcal{M}^{-1} |\). Then the following propositions holds: {\parindent=6mm \begin{itemize} \item[1).] Suppose that \(k>r\). Then the following conditions are equivalent:NEWLINENEWLINE\item[a)] the slope inequality \((\star)\) holds; b) \(K^{r}_{f} \geq 0\) and \(\text{deg} \, f_{*} \omega_{f} \geq 0\); c) \(y/k \leq \mu\). \item [2).] If \(k > 1\), then the line bundle \(\mathcal{O}_{X}(h)\) is \(f\)-positive for any \(h\geq 1\) if and only if \(y/k \leq \mu\). NEWLINENEWLINE\end{itemize}} In the further part of the paper, the authors investigate the meaning of Theorem 1 on the geometry of \(X\) and of its fibers. Some of these results are devoted to instability and singularity conditions on the fibers and also the total space of \(X\). We recall here only some of them. Let us consider the Harder-Narasimhan filtration of \(\mathcal{E}\): NEWLINE\[NEWLINE0 = \mathcal{E}_{0} \subset \mathcal{E}_{1} \subset \dots \subset \mathcal{E}_{\ell} = \mathcal{E},NEWLINE\]NEWLINE and call \(\mu_{i} :=\mu( \mathcal{E}_{i} / \mathcal{E}_{i-1})\). Recall in particular that NEWLINE\[NEWLINE\mu_{\ell} < \mu_{ \ell - 1} < \dots < \mu_{1},NEWLINE\]NEWLINE and \(\mu_{\ell} < \mu < \mu_{1}\) unless \(\mathcal{E}\) is semistable, in which case \(\mathcal{E}_{1} = \mathcal{E}_{\ell} = \mathcal{E}\). Let \(Y\) be a subvariety of dimension \(s\) and of degree \(k\) in \(\mathbb{P}^{n}\). Consider the set \(Z(Y)\) of all \((n-s-1)\)-dimensional projective subspaces \(L\) in \(\mathbb{P}^{n}\) intersecting \(Y\). This is an hypersurface of degree \(k\) in the Grassmannian \(G:= \text{Gr}(n-s,n+1)\) called the Chow variety. It is defined by the vanishing of some polynomial of degree \(k\) in the Grassmann coordinate ring (up to a unique constant factor) and this element is called the Chow form of \(Y\). The variety \(Y\) is called Chow semistable (resp. stable) if it its Chow form is semistable (resp. stable) for the natural \(\mathrm{SL}(n+1)\)-action.NEWLINENEWLINE{ Theorem 2.} Let \(\mathcal{E}\) be a \(\mu\)-unstable sheaf. Then, given any relative hypersurface \(X \in | \mathcal{O}_{\mathbb{P}}(k) \otimes \mathcal{M}^{-1}|\) with \(y/k > \mu\), any fiber \(f : X \rightarrow B\) is Chow unstable with respect to \(\mathcal{O}_{F}(h)\) for any \(h \geq 1\).NEWLINENEWLINEUsing this result the authors proved a condition on singularities of the effective divisors in \(| \mathcal{O}_{\mathbb{P}}(k) \otimes \mathcal{M}^{-1}|\) with \(y/k \geq \mu\), which involves the log canonical threshold of a pair \((\Sigma, X|_{\Sigma})\) with \(\Sigma\) general fiber of \(\pi\).NEWLINENEWLINE{ Theorem 3.} Let \(X\) be a relative hypersurface from \(| \mathcal{O}_{\mathbb{P}}(k) \otimes \mathcal{M}^{-1}|\). If \(y/k \in ( \mu, \mu_{1}]\), then any fiber \(F\) of \(f\) is singular and NEWLINE\[NEWLINE\text{lct}(\Sigma, X|_{\Sigma}) < \frac{r}{k}.NEWLINE\]NEWLINENEWLINENEWLINEAnother result provides some conclusions on singularities of \(X\).NEWLINENEWLINE{ Theorem 4.} If \(X \in | \mathcal{O}_{\mathbb{P}}(k) \otimes \mathcal{M}^{-1}|\) is smooth or is such that \(\text{lct}(\mathbb{P},X) \geq r/k\), then \(y/k \leq \mu\).NEWLINENEWLINEIn the last part of the paper the authors study, using the Nakayama-Zariski decomposition, the schematic fixed locus of any divisor under the additional assumption that \(\ell = 2\). Let us recall the main result of this part.NEWLINENEWLINE{ Theorem 5.} Suppose that \(\ell = 2\) and consider the linear system \(|\mathcal{O}_{\mathbb{P}}(km) \otimes \mathcal{M}^{-m}|\). Suppose that \(y/k > \mu_{2}\) (the system is big and not nef) and that \((\mu_{1} - \mu_{2})\) divides \((y-\mu_{2}k)\). Then the fundamental cycle of the schematic fixed locus of \(|\mathcal{O}_{\mathbb{P}}(km) \otimes \mathcal{M}^{-m}|\) contains the codimension \(\text{rank} \, \mathcal{E}_{1}\) cycle NEWLINE\[NEWLINEm \frac{y-\mu_{2}k}{\mu_{1} - \mu_{2}}\mathbb{P}(\mathcal{E}/\mathcal{E}_{1})NEWLINE\]NEWLINE and asymptotically coincides with it.