In this chapter, we study the codes of nodal quintic surfaces. The most interesting situation is when the quintic has an even set of 20 nodes. We prove that in such a case the quintic can have at most 28 nodes. It follows that its associated code is a shortening of one of exactly two codes. One has length 26 and the other has length 28. We show that for length 28 we obtain an irreducible component of the variety of quintics with 28 nodes, which can be described as the discriminant of a special web of 3-dimensional quadrics. If there is no even set of 20 nodes, then the code is a shortening of the biduality code. We also discuss some open questions, as the existence of quintics corresponding to shortenings of the code of length 26, and the determination of the fundamental group of the smooth locus.

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Nodal Quintic Surfaces

  • Fabrizio Catanese,
  • Alessandro Verra

摘要

In this chapter, we study the codes of nodal quintic surfaces. The most interesting situation is when the quintic has an even set of 20 nodes. We prove that in such a case the quintic can have at most 28 nodes. It follows that its associated code is a shortening of one of exactly two codes. One has length 26 and the other has length 28. We show that for length 28 we obtain an irreducible component of the variety of quintics with 28 nodes, which can be described as the discriminant of a special web of 3-dimensional quadrics. If there is no even set of 20 nodes, then the code is a shortening of the biduality code. We also discuss some open questions, as the existence of quintics corresponding to shortenings of the code of length 26, and the determination of the fundamental group of the smooth locus.