Regular del Pezzo surfaces with irregularity (Q2804211)

The author answers affirmatively the question of whether there exist regular del Pezzo surfaces \(X\) that are geometrically non-normal or geometrically non-reduced and have positive irregularity \(h^{1}(\mathcal{O}_{X}) >0\). Before we formulate the main result of the paper let us recall some notions.NEWLINENEWLINEA \(k\)-variety \(X\) is said to satisfy a property geometrically if the base change \(X_{\bar{k}}\) to the algebraic closure satisfies the given property.NEWLINENEWLINEA variety \(X\) is defined to be regular provided that the local coordinate ring \(\mathcal{O}_{X,x}\) is a regular local ring at all points \(x \in X\). A \(k\)-variety \(X\) is smooth over \(k\) provided that it is geometrically regular.NEWLINENEWLINEA \textit{del Pezzo} surface over a field \(k\) is defined to be a \(2\)-dimensional projective variety \(X\) over \(k = H^{0}(X, \mathcal{O}_{X})\) which is \textit{Fano}, that is, a Gorenstein scheme for which the inverse of the dualizing sheaf \(\omega_{X}^{-1}\) is an ample line bundle.NEWLINENEWLINE{Main Result.} {\parindent=6mm \begin{itemize} \item[1)] There exist regular del Pezzo surfaces \(X_{1}\) and \(X_{2}\) with irregularity \(h^{1}(\mathcal{O}_{X_{i}}) = 1\) and of degrees \(K_{X_{1}}^{2} = 1\) and \(K_{X_{2}}^{2} = 2\). The surface \(X_{1}\) is geometrically integral and defined over the field \(\mathbb{F}_{2}(\alpha_{0},\alpha_{1}, \alpha_{2})\), while \(X_{2}\) is geometrically non-reduced and defined over the index-\(2\) subfield \(\mathbb{F}_{2}(\alpha_{i}\alpha_{j} : 0 \leq i,j \leq 3) \subset \mathbb{F}_{2}(\alpha_{0}, \alpha_{1}, \alpha_{2}, \alpha_{3})\). \item [2)] If \(X\) is a normal del Pezzo surface (e.g. a regular del Pezzo surface) with irregularity \(q>0\) and anti-canonical degree \(d = K_{X}^{2}\) over a field of characteristic \(p\), then NEWLINE\[NEWLINEq \geq \frac{d(p^{2}-1)}{6}.NEWLINE\]NEWLINE NEWLINENEWLINE\end{itemize}} It is worth pointing out that the proof of \(1)\) is constructive, i.e. in Section 3, the author constructs surfaces \(X_{1}\) and \(X_{2}\) explicitly. Moreover, in Section 4. the author presents concrete description of the geometry in the case of \(X_{1}\).NEWLINENEWLINE{ Proposition. } There exists a regular form \(Z\) of a double plane in \(\mathbb{P}^{3}\) and a finite, inseparable morphism \(f: Z \rightarrow X\) of degree \(p=2\). Moreover, if \(\bar{Z}\) and \(\bar{X_{1}}\) denote the geometric base change of \(Z\) and \(X_{1}\), respectively, then this construction has the following properties: {\parindent=6mm \begin{itemize} \item[a)] The induced morphism \(f^{\mathrm{red}} : \mathbb{P}^{2} \cong \bar{Z}^{\mathrm{red}} \rightarrow \bar{X_{1}}\) is the normalization of \(\bar{X_{1}}\). \item [b)] The singular locus of \(\bar{X_{1}}\) is a rational cuspidal curve \(C\) of arithmetic genus one. \item [c)] The inverse image of \(C\) under \(f^{\mathrm{red}}\) is a non-reduced double line in \(\mathbb{P}^{2}\). NEWLINENEWLINE\end{itemize}} In order to obtain the mentioned results the author studies inseparable degree \(p\) covers associated to Frobenius-killed classes in the first cohomology group of pluri-canonical line bundles (which were studied by \textit{T. Ekedahl} [Proc. Symp. Pure Math. 46, 139--149 (1987; Zbl 0659.14018)]) and algebraic foliations on regular varieties (which extends some results of \textit{T. Ekedahl} [Publ. Math., Inst. Hautes Étud. Sci. 67, 97--144 (1988; Zbl 0674.14028)] from the smooth case).