六次曲面上的偶节点集
Even sets of nodes on sextic surfaces
摘要：
我们确定可能的甚至设置的节点在六次曲面上的$\Pn 3$，特别显示，他们的基数正好是在一组$\{24, 32, 40, 56 \}$的数字。我们还表明，所有可能的情况下承认一个明确的描述。我们使用的方法是一个相互作用的编码理论和射影几何一方面，同调和计算机代数的其他。我们给出了一个详细的尺规作图法的新情况下，甚至一组56个节点，但存在的最终验证依赖于计算机计算。该结构给出了一个最大的家庭，unirational和27维，节点六次线，甚至设置56个节点。根据Casnati和Catanese的一个一般定理，每个这样的节面$F$被给出为适当向量丛$\E$依赖于$F$的对称映射的行列式.这里的第一个困难是证明这种向量丛的存在性。这导致我们调查迄今未知的模空间秩6向量丛，我们证明是双有理的一个模空间的平面表示的三次曲面$\PP^3$。由此产生的图片显示了一个非常丰富和有趣的几何。主要的困难是要显示存在的“好”地图$ϕ$，和有趣的现象，显示了以下内容：“模空间“的这种对$(\E, ϕ)$是可减少的，事实上，对于一般选择的$\E$的行列式的$ ϕ$是双三次曲面$G$。
英文摘要：
We determine the possible even sets of nodes on sextic surfaces in $\Pn 3$, showing in particular that their cardinalities are exactly the numbers in the set $\{24, 32, 40, 56 \}$. We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, of homological and computer algebra on the other. We give a detailed geometric construction for the new case of an even set of 56 nodes, but the ultimate verification of existence relies on computer calculations. The construction gives a maximal family, unirational and of dimension 27, of nodal sextics with an even set of 56 nodes. According to a general theorem of Casnati and Catanese each such nodal surface $F$ is given as the determinant of a symmetric map for an appropriate vector bundle $\E$ depending on $F$. The first difficulty here is to show the existence of such vector bundles. This leads us to the investigation of a hitherto unknown moduli space of rank 6 vector bundles which we show to be birational to a moduli space of plane representations of cubic surfaces in $\PP^3$. The resulting picture shows a very rich and interesting geometry. The main difficulty is to show the existence of "good" maps $ϕ$, and the interesting phenomenon which shows up is the following: the "moduli space" of such pairs $(\E, ϕ)$ is reducible, indeed for a general choice of $\E$ the determinant of $ ϕ$ is the double of a cubic surface $G$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
