부산대학교 도서관

The list of conjugacy classes of finite subgroups of the projective linear group $PGL(2,{\Bbb C})$ has been known since the 19th century. The only simple group in this list is the group $A_5$ of symmetries of the icosahedron inscribed in the Riemann sphere. The analog of the group $PGL(2,{\Bbb C})$ in higher dimension is the Cremona group $Cr_n({\Bbb C})$ of birational transformations of the projective space ${\Bbb P}^n$ of dimension $n$. The icosahedral group also appears in the classification of simple finite subgroups of this group that has been known only in the cases $n = 2$ [I.~V. Dolgachev and V.~A. Iskovskikh, in {\it Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I}, 443--548, Progr. Math., 269, Birkhäuser Boston, Inc., Boston, MA, 2009; MR2641179] and $n = 3$ [Y.~G. Prokhorov, J. Algebraic Geom. {\bf 21} (2012), no.~3, 563--600; MR2914804]. The classification of conjugacy classes of such groups has been known only in the case $n = 2$. They are realized as groups of projective transformations of the plane, a quadric surface, or a del Pezzo surface of degree 5. The main motivation of the book under review is to extend this classification to the case $n = 3$. However, the content of the book is more modest and restricts to an extensive study of only one conjugacy class realized as the group of projective symmetries of the rational Fano threefold $V_5$ of degree 5 in ${\Bbb P}^6$. The authors prove, for example, that such a variety is $A_5$-rigid, i.e., is not $A_5$-equivariantly isomorphic to any other 3-dimensional Mori space (a Fano threefold, or a del Pezzo fibration, or a conic bundle). To prove this result, one has to discuss some new techniques of birational geometry like, for example, the theory of Sarkisov links; this partially explains the large size of the book. The authors also include some nice geometrical properties (not used in the proof of the rigidity) of the variety $V_5$ related to its group of symmetries. \par Many occurrences of the icosahedron group as the group of symmetries of various algebraic varieties have been known for a long time (e.g.\ the Clebsch diagonal cubic surfaces, the Bring curve, the quintic del Pezzo surface). The authors add to the list some new examples. For example, they classify quartic surfaces in ${\Bbb P}^3$ admitting $A_5$ as a group of projective symmetries. This was used in a recent paper of the authors and V. Przyjalkowski [Eur. J. Math. {\bf 2} (2016), no.~1, 96--119; MR3454093] that discussed other conjugacy classes of $A_5$ in $Cr_3({\Bbb C})$ (realized by double covers of ${\Bbb P}^3$ branched along quartic surfaces with $A_4$-symmetry). Another interesting example discussed with great detail is the unique $A_5$-invariant smooth complete intersection of $V_5$ with a cubic hypersurface in ${\Bbb P}^6$. \par The reviewer readily agrees with the authors' belief that the book `can provide the same kind of aesthetic feeling as one that possibly stood behind the classical works of Klein, Maschke, Blichfeldt and many others regarding symmetry groups'.