Cubic surfaces and their invariants: Some memories of Raymond Stora

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• 作者：Michel Bauer ^{michel.bauer@cea.fr" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Nuclear Physics B
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：912
• 期：Complete
• 页码：374-425
• 全文大小：860 K

文摘

Cubic surfaces embedded in complex projective 3-space are a classical illustration of the use of old and new methods in algebraic geometry. Recently, they made their appearance in physics, and in particular aroused the interest of Raymond Stora, to the memory of whom these notes are dedicated, and to whom I'm very much indebted.

Each smooth cubic surface has a rich geometric structure, which I review briefly, with emphasis on the 27 lines and the combinatorics of their intersections. Only elementary methods are used, relying on first order perturbation/deformation theory.

I then turn to the study of the family of cubic surfaces. They depend on 20 parameters, and the action of the 15 parameter group SL₄(C)$S{L}_4(\mathbb{C})$ splits the family in orbits depending on 5 parameters. This takes us into the realm of (geometric) invariant theory. I review briefly the classical theorems on the structure of the ring of polynomial invariants and illustrate its many facets by looking at a simple example, before turning to the already involved case of cubic surfaces. The invariant ring was described in the 19th$19\text{th}$ century. I show how to retrieve this description via counting/generating functions and character formulae.