论文标题
淋巴结表面的品种，编码理论和立方曲面的判别因子。第1部分：一般性和节点K3表面。第2部分：立方曲面，相关的判别因子。第3部分：节点五精灵。第4部分：节点六元
Varieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics
论文作者
论文摘要
我们将两个二进制代码附加到一个投射节点表面（严格的代码k，甚至对于D度d，扩展代码k'），以调查D度d的p^3中的鼻表面nodal severi品种f（d，n）f（d，n）。我们的第一个主要结果解决了一个可以追溯到100年来的问题：F（4，n）的不可约合的组成部分与它们扩展代码K'的同构类别进行了两者的态度，并且这些完全是扩展的Kummer代码K'的34个可能的缩写，并且仅在另一个代码的范围内，并且只有另一个代码的代码，则是代码的代码。我们将此结果扩展到以相同方式分类所有节点K3表面的不可还原组件，并将其完全分类为它们的扩展代码。在此分类中，有一些零星的情况，通过节点通过投影获得。 对于p^3中D = 5的表面，我们确定（一个可能的例外）所有可能的代码k，对于K的几种情况，我们显示了相应的f（5，n）的相应开放式集合的不可约性，例如，我们显示了togliatti Quintic Quintic表面的不可约性。在第四部分中，我们表明，“ togliatti like”描述适用于6度的表面，最大节点= 65：它们是p^6中有31个（分别为32）个节点的Cubic Hypersurfaces的判别，并且我们有一个不可估的18维家族。对于D度= 6，我们的主要结果是基于一些新颖的辅助结果：1）研究六分布的半个节点集的研究，2）对三次超曲面的判别因子的研究x，3）计算机辅助证明，对于n = 65，两个代码k，k，k，k，k，k，k，k，k，k，k，k，k's的描述都具有独特的范围，4）的描述是相关的。 Doro-hall图。
We attach two binary codes to a projective nodal surface (the strict code K and, for even degree d, the extended code K' ) to investigate the `Nodal Severi varieties F(d, n) of nodal surfaces in P^3 of degree d and with n nodes, and their incidence hierarchy, relating partial smoothings to code shortenings. Our first main result solves a question which dates back over 100 years: the irreducible components of F(4, n) are in bijection with the isomorphism classes of their extended codes K', and these are exactly all the 34 possible shortenings of the extended Kummer code K' , and a component is in the closure of another if and only if the code of the latter is a shortening of the code of the former. We extend this result classifying the irreducible components of all nodal K3 surfaces in the same way, and we fully classify their extended codes. In this classification there are some sporadic cases, obtain through projection from a node. For surfaces of degree d=5 in P^3 we determine (with one possible exception) all the possible codes K, and for several cases of K, we show the irreducibility of the corresponding open set of F(5, n), for instance we show the irreducibility of the family of Togliatti quintic surfaces. In the fourth part we show that a `Togliatti-like' description holds for surfaces of degree 6 with the maximum number of nodes= 65: they are discriminants of cubic hypersurfaces in P^6 with 31 (respectively 32) nodes, and we have an irreducible 18-dimensional family of them. For degree d=6, our main result is based on some novel auxiliary results: 1) the study of the half-even sets of nodes on sextic surfaces, 2) the investigation of discriminants of cubic hypersurfaces X, 3) the computer assisted proof that, for n = 65, both codes K, K' are uniquely determined, 4) the description of these codes, relating the geometry of the Barth sextic with the Doro-Hall graph.
