Notes on canonical surfaces
Notes on canonical surfaces
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DOI：
10.2748/tmj/1178227542
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发表时间：
1991-03
影响因子：
0.5
通讯作者：
E. Horikawa
中科院分区：
文献类型：
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作者：
E. Horikawa
极小代数曲面S称为标准曲面，如果与标准系相联系的映射Φκ：S-+P，n=Pg-，在其像上诱导出S的二元映射。设四次曲面(S)通过像ΦK(S)表示所有二次曲面的交。如果S是标准曲面，则c>；2>；Pg-L(见[10，第二部分，引理1.1])。如果这里的等号成立，S的结构相当简单，其构造可以完全如文[2]所述。这些本质上都归功于Castelnuovo[4]，我在1976年获得了我的证明，这基本上类似于[2，§1-4]。此外，我注意到一些具有Pg=L的正则曲面(使得四边形(S)是凡尔纳曲面上的锥体)阻碍了变形。对于这样的S，|K|是不充分的，并且正则系统|Kt|对S的任何小变形ST都是不充分的，因此，在[3]中，S具有一般的非约化模。这在[10，第三部分，关于第229页的注解]中被暗示，但有一个错误的引文pg-β，c\=11。(我计划写一篇题为《关于某些典型曲面》的论文，与c\=3pg-L和3pg-β一起讨论曲面，但一直没有完成。)这个表面是米兰达最近独立发现的。但他忽略了一点：如果{ST：Tem}是参数空间M上的平坦族，那么Tem}是否形成平坦族？通常情况下并非如此，因为在某些情况下，的维度可能会跳跃。
A minimal algebraic surface S is called a canonical surface if the map Φκ: S-+P , n=pg—\, associated to the canonical system \K\ induces a birational map of S onto its image. Let Quad(S) denote the intersection of all the quadrics through the image ΦK(S). If S is a canonical surface, then c\>2>pg — l (see [10, Part II, Lemma 1.1]). If the equality sign holds here, S has rather simple structure and its construction can be completely described as in [2]. These are all essentially due to Castelnuovo [4], and I obtained my proof in 1976, which is mostly similar to [2, §§1-4]. Moreover, I noticed that some of the canonical surfaces with pg = l, c\ = \<\ (such that Quad(S) is a cone over the Veronese surface) have obstructed deformations. For such S, | K | is not ample, and the canonical system | Kt | remains non-ample for any small deformation St of S. So, by [3], S has generically non-reduced moduli. This was insinuated in [10, Part III, Remark on p. 229], but with an erroneous citation pg — β, c\ = 11. (I planned to write a paper entitled "On certain canonical surfaces" to discuss surfaces with c\ = 3pg — l and 3pg — β, but it was never completed.) This surface was independently found recently by Miranda [15]. But he missed one point: If {St: teM} is a flat family over a parameter space M, then does teM} form a flat family? This is not true in general, because the dimension of may jump in some case.