부산대학교 도서관

A {\it Beauville surface (of unmixed type)} is a complex algebraic surface $S$ such that \roster \item $S$ is isogenous to a {\it higher product}, that is, $S\cong (C_1 \times C_2)/G$, where $C_1$ and $C_2$ are algebraic curves of genera at least 2 and $G$ is a finite group acting freely on ${C_1 \times C_2}$; \item $G$ acts faithfully as a group of automorphisms of each $C_i$ so that $C_i/G \cong \Bbb{P}^1$ and the covering map $C_i \to C_i/G$ is ramified over three points. \endroster In his well-known textbook on complex algebraic surfaces [{\it Surfaces algébriques complexes}, Soc. Math. France, Paris, 1978; MR0485887; {\it Complex algebraic surfaces}, translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid, second edition, London Math. Soc. Stud. Texts, 34, Cambridge Univ. Press, Cambridge, 1996; MR1406314], A. Beauville gave such an example by considering a certain finite quotient of the self-product of the Fermat curve defined by $x^n+y^n+z^n=0$. Later, inspired by Beauville's example, F. Catanese formally introduced this notion in his study of finite quotients of products of curves and the related moduli problem of surfaces of general type [see Amer. J. Math. {\bf 122} (2000), no.~1, 1--44; MR1737256]. \par Surprisingly, it turns out that the above geometric notion can be completely characterized in terms of the group structure of $G$, the so-called {\it Beauville structure}. Hence, the classification of all Beauville surfaces becomes a group-theoretic problem. \par Recall that given a finite group $G$, a {\it Beauville structure} for $G$ is a set of pairs of elements $\{ \{x_1, y_1\}, \{x_2, y_2\}\} \subset G \times G$ such that $\langle x_1, y_1 \rangle = \langle x_2, y_2 \rangle = G$ and $$ \Sigma(x_1, y_1) \cap \Sigma(x_2, y_2) = \{ e_G \}, $$ where $\Sigma(x_i, y_i)$ denotes the set of all conjugates of the powers of $x_i$, $y_i$ and $x_iy_i$ in $G$. We then call $G$ a {\it Beauville group} if it admits a Beauville structure. Note that this definition is slightly different from that in the literature where certain generating triples of $G$ are involved. However, as one can see, Beauville groups are actually generated by two elements. \par The classification of abelian, nilpotent, (quasi)simple and strongly real Beauville groups has attracted a lot of attention since it was introduced by Catanese [see I.~C. Bauer, F. Catanese and F.~J. Grunewald, in {\it Geometric methods in algebra and number theory}, 1--42, Progr. Math., 235, Birkhäuser Boston, Boston, MA, 2005; MR2159375; Y. Fuertes, G. González~Díez and A. Jaikin-Zapirain, Groups Geom. Dyn. {\bf 5} (2011), no.~1, 107--119; MR2763780; S. Garion, M.~J. Larsen and A. Lubotzky, J. Reine Angew. Math. {\bf 666} (2012), 225--243; MR2920887; R.~M. Guralnick and G. Malle, J. Lond. Math. Soc. (2) {\bf 85} (2012), no.~3, 694--721; MR2927804; B. Fairbairn, K. Magaard and C.~W. Parker, Proc. Lond. Math. Soc. (3) {\bf 107} (2013), no.~4, 744--798; MR3108830; B. Fairbairn, in {\it Groups St Andrews 2013}, 225--241, London Math. Soc. Lecture Note Ser., 422, Cambridge Univ. Press, Cambridge, 2015; MR3495658]. Also, Beauville surfaces have been used to construct interesting orbits of the absolute Galois group $\roman{Gal}(\overline{\Bbb{Q}}/\Bbb{Q})$ [G. González~Díez and A. Jaikin-Zapirain, Proc. Lond. Math. Soc. (3) {\bf 111} (2015), no.~4, 775--796; MR3407184]. Furthermore, Beauville's original example has also been used by S.~S. Galkin and E. Shinder [Adv. Math. {\bf 244} (2013), 1033--1050; MR3077896] to construct examples of exceptional collections of line bundles. \par The author surveys the very recent advances on Beauville $p$-groups, including constructions of (non)abelian Beauville $p$-groups and their structural properties. Several open questions, problems and conjectures are also discussed at the end.