Abstract
It is known that every Fano–Mukai fourfold X of genus 10 is acted upon by an involution \(\tau \) which comes from the center of the Weyl group of the simple algebraic group of type \(\textrm{G}_2\), see Prokhorov and Zaidenberg (Eur J Math 4(3):1197–1263 2018, Eur J Math 8:561–572, 2022). This involution is uniquely defined up to conjugation in the group \({\text {{Aut}}}(X)\). In this note we describe the set of fixed points of \(\tau \) and the surface scroll swept out by the \(\tau \)-invariant lines.
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Notes
The upper index \(\textrm{sr}\) in \(\Omega ^{\textrm{sr}}\) stands for “subregular”.
Given two algebraic cycles X, Y on a smooth variety W, we write \(X\cdot Y\) for the scheme-theoretic intersection of X and Y, while \(X\cap Y\) stands for the reduced intersection of subvarieties. By abuse of notation, in the case where X and Y are cycles of complementary dimensions, \(X\cdot Y\) can also stand for the intersection number; in such cases, the meaning is clear from the context.
Alternatively, under the assumption that \(E_1=E_2\) it can be shown that \(s|_{{\tilde{\Pi }}}:{\tilde{\Pi }}\rightarrow \Pi \) sends some pair \((p_1,p_2)\) of distinct points \(p_i\in {\tilde{E}}_i\cap {\tilde{r}}\), \(i=1,2\) on the same ruling \({\tilde{r}}\) of \({\tilde{\Pi }}\) to the same point of E. Indeed, \(\iota :{\tilde{r}}\mapsto (p_1,p_2)\) embeds C in \(E_1\times E_2\), and \(s|_{E_1\cup E_2}\) induces a morphism \(\phi :E_1\times E_2\rightarrow E\times E\). The curve \(\phi \circ \iota (C)\) on \(E\times E\) intersects the diagonal \(\delta \) of \(E\times E\), which proves our claim. However, we know already that \(r=s({\tilde{r}})\) is a ruling of \(\Pi \) and \(s|_{{\tilde{r}}}:{\tilde{r}}\rightarrow r\) is an isomorphism. This yields the desired contradiction.
Cf. also [22, Lemma 2.11(ii)–(iii)] for a particular case where S is a cubic cone.
Notice that \({\tilde{\gamma }}_S\ne {\tilde{E}}_i\) for a general \(S\in {\mathscr {S}}_1\), \(i=1,2\). Indeed, otherwise \(E_i\subset A_S\cap \tau (A_S)\) for any \(S\in {\mathscr {S}}_1\) contrary to Proposition 1.3(l).
Cf. also [6, Theorem 4.5].
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Acknowledgements
This paper is based on joint articles [20,21,22] of Yuri Prokhorov and the author. We are grateful to Yuri Prokhorov for useful discussions; the ideas of several proofs are due to him. Our thanks also due to Ciro Ciliberto for his kind assistance, and especially for producing several amazing examples of singular surface scrolls. We are grateful to the anonymous referee, whose comments were relevant and particularly helpful in avoiding inaccuracies, clarifying arguments and improving style.
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In memory of Jean-Pierre Demailly.
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Zaidenberg, M. Central Weyl involutions on Fano–Mukai fourfolds of genus 10. Rend. Circ. Mat. Palermo, II. Ser 72, 3277–3303 (2023). https://doi.org/10.1007/s12215-023-00936-x
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DOI: https://doi.org/10.1007/s12215-023-00936-x