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Exceptional groups and del Pezzo surfaces. (English) Zbl 1080.14533

Bertram, Aaron (ed.) et al., Symposium in honor of C. H. Clemens. A weekend of algebraic geometry in celebration of Herb Clemens’s 60th birthday, University of Utah, Salt Lake City, UT, USA, March 10–12, 2000. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2152-0/pbk). Contemp. Math. 312, 101-115 (2002).
From the introduction: Let \({\mathbf E}_r\), \(r=6,7,8\) denote the simply connected form of the complex linear group whose root system is of type \(E_r\). We extend this series to \(3\leq r\leq 8\) by setting \({\mathbf E}_5=D_5\), \(E_4=A_4\), and \(E_3=A_1\times A_2\), again with the understanding that \({\mathbf E}_r\) denotes the simply connected form of the corresponding complex linear group of type \(E_r\). It has long been known that there are deep connections between \({\mathbf E}_r\) and del Pezzo surfaces of degree \(9-r\). The goal of this paper is to make one of these connections explicit: if \(X\) is a del Pezzo surface of degree \(d=9-r\), possibly with rational double point singularities, we show that there is a “tautological” holomorphic \(\widetilde{\mathbf E}_r\)-bundle \(\Xi\) over \(X\), where \(\widetilde {\mathbf E}_r\) is an appropriate conformal form of the group \({\mathbf E}_r\). The most classical case, \(r=6\), corresponds to the case of cubic surfaces. In this case, \[ \widetilde{\mathbf E}_6={\mathbf E}_6 \times_{\mathbb{Z}/3\mathbb{Z}}\mathbb{C}^*. \] There is a natural 27-dimensional representation \(\rho\) of \({\mathbf E}_6\) and \(\widetilde {\mathbf E}_6\). If \(X\) is a smooth cubic surface, then the induced holomorphic vector bundle \(\Xi\times_{\widetilde{\mathbf E}_6}\mathbb{C}^{27}\) is isomorphic to \(\bigoplus^{27}_{i=1}{\mathcal O}_X(L_i)\), where the \(L_i\) are the distinct lines on \(X\). The fact that \(\Xi\times_{\widetilde{\mathbf E}_6} \mathbb{C}^{27}\) is isomorphic to a direct sum of line bundles reflects the fact that the structure group of \(\Xi\) reduces to a maximal torus of \(\widetilde{\mathbf E}_6\). When \(X\) has rational double points, the induced rank 27 vector bundle is no longer a direct sum of line bundles. Instead, the line bundle factors on a general surface coalesce into irreducible summands of higher rank, reflecting the way in which lines coalesce on singular cubic surfaces. Correspondingly, the structure group of \(\Xi\) reduces to a reductive subgroup of \(\widetilde{\mathbf E}_6\) whose Lie algebra is generated by a maximal torus and by roots corresponding to smooth rational curves of self-intersection \(-2\) in the minimal resolution of \(X\). This phenomenon reflects the picture ia physics, where rational double point singularities correspond to extra massless particles, and these particles are described as gauge particles for a gauge group formed in exactly the same way. Similar results hold for any value of \(r\). One motivation for describing the bundle \(\Xi\) is to give a more direct explanation for the correspondence described in [R. Friedman, J. W. Morgan and E. Written, Math. Res. Lett. 5, 97–118 (1998; Zbl 0937.14019)] between \(S\)-equivalence classes of semistable between \(S\)-equivalence classes of semistable \({\mathbf E}_r\)-bundles over a smooth elliptic curve \(E\) with origin \(p_0\) and triples \((X,D,\varphi)\), where \(X\) is a del Pezzo surface of degree \(9-r\), \(D\) is a hyperplane section of \(X\), and \(\varphi:D\to E\) is an isomorphism from \(D\) to \(E\) such that \(\varphi^*{\mathcal O}_E((9-r)p_0)\cong{\mathcal O}_X (D)|D\). We show that, given the triple \((X,D,\varphi)\), after a suitable twist by a line bundle, there is a canonical reduction of the structure group of the bundle \(\varphi^*(\Xi|D)\) to an \({\mathbf E}_r\)-bundle \(\xi\) over \(E\) which realizes the above correspondence. Moreover, in our construction, the bundle \(\xi\) is always the regular representative. Moreover, in section 5 we discuss briefly the relation of the numerology of the lines, conics etc. to the weights of the fundamental representations of \(\mathbb{E}_7\).
For the entire collection see [Zbl 1002.00010].

MSC:

14J25 Special surfaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14L35 Classical groups (algebro-geometric aspects)

Citations:

Zbl 0937.14019
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