In our previous work [4] we showed that the extension property holds for linear codes over a finite module alphabet with respect to the symmetrized weight composition if and only if the socle of the alphabet is cyclic. For alphabets with a noncyclic socle we proved the existence of an unextendable isometry. However, our construction was implicit. This paper deals with a particular case of an al...

We formulate and prove a generalization of Zariski-van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz-Zariski-van Kampen type for the fundamental groups of the complements to the Grassmannian dual varieties.

4 Zariski sheaves associated with pretheories. 25 4.1 Technical lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Pretheories in a neighborhood of a smooth subscheme. . . . . . 31 4.3 Pretheories on curves over a field. . . . . . . . . . . . . . . . . 35 4.4 Pretheories on semi-local schemes. . . . . . . . . . . . . . . . . 36 4.5 Zariski cohomology of sheaves associated with pre...

The strength of a homogeneous polynomial (or form) is the smallest length an additive decomposition expressing it whose summands are reducible forms. Using functors, we show that set forms with bounded not always Zariski-closed. More specifically, if ground field algebraically closed, prove quartics ≤3 Zariski-closed for large number variables.

We describe the current state of progress on the maximal subgroup problem for the Monster sporadic simple group. Any unknown maximal subgroup is an almost simple group whose socle is in one of 19 specified isomorphism classes.