Dimension counts for singular rational curves.
Cotterill, Ethan
Dimension counts for singular rational curves. - Rio de Janeiro: IMPA, 2015. - video online
Talk.
Rational curves are essential tools for classifying algebraic varieties. Establishing dimension bounds for families of embedded rational curves that admit singularities of a particular type arises arises naturally as part of this classification. Singularities, in turn, are classified by their value semigroups. Unibranch singularities are associated to numerical semigroups, i.e. subsemigroups of the natural numbers. These fit naturally into a tree, and each is associated with a particular weight, from which a bound on the dimension of the corresponding stratum in the Grassmannian may be derived. Understanding how weights grow as a function of (arithmetic) genus g, i.e. within the tree, is thus fundamental. We establish that for genus g \leq 8, the dimension of unibranch singularities is as one would naively expect. Multibranch singularities are far more complicated; in this case, we give a general classification strategy and again show, using semigroups, that dimension grows as expected relative to g when g \leq 5. This is joint work with Lia Fusaro Abrantes and Renato Vidal Martins .
Matematica.
Dimension counts for singular rational curves. - Rio de Janeiro: IMPA, 2015. - video online
Talk.
Rational curves are essential tools for classifying algebraic varieties. Establishing dimension bounds for families of embedded rational curves that admit singularities of a particular type arises arises naturally as part of this classification. Singularities, in turn, are classified by their value semigroups. Unibranch singularities are associated to numerical semigroups, i.e. subsemigroups of the natural numbers. These fit naturally into a tree, and each is associated with a particular weight, from which a bound on the dimension of the corresponding stratum in the Grassmannian may be derived. Understanding how weights grow as a function of (arithmetic) genus g, i.e. within the tree, is thus fundamental. We establish that for genus g \leq 8, the dimension of unibranch singularities is as one would naively expect. Multibranch singularities are far more complicated; in this case, we give a general classification strategy and again show, using semigroups, that dimension grows as expected relative to g when g \leq 5. This is joint work with Lia Fusaro Abrantes and Renato Vidal Martins .
Matematica.