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Curves in projective space and RSK

Carl Lian, Saskia Solotko

Abstract

The geometric Tevelev degrees of projective space enumerate general, pointed algebraic curves interpolating through the maximal possible number of points. Previous work expresses these invariants in terms of Schubert calculus. Extending ideas of Gillespie--Reimer-Berg, we use the RSK correspondence to give a positive interpretation of these counts in terms of the combinatorics of words.

Curves in projective space and RSK

Abstract

The geometric Tevelev degrees of projective space enumerate general, pointed algebraic curves interpolating through the maximal possible number of points. Previous work expresses these invariants in terms of Schubert calculus. Extending ideas of Gillespie--Reimer-Berg, we use the RSK correspondence to give a positive interpretation of these counts in terms of the combinatorics of words.

Paper Structure

This paper contains 8 sections, 14 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Geometric results
  3. Combinatorial results
  4. Acknowledgments
  5. Preliminaries
  6. Operations on the red and blue tableaux
  7. RSK
  8. Remarks on words

Key Result

Theorem 1.1

Assume eq:dim_constraint. Then, $\mathop{\mathrm{Tev}}\nolimits^{\mathbb{P}^r}_{g,n,d}$ is equal to the number of $(r+1)$-ary words of length $g$ (Definition def:word) which have:

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Theorem 2.6
  • ...and 36 more