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Two-pointed Prym-Brill-Noether Loci and coupled Prym-Petri theorem

Minyoung Jeon

Abstract

We establish two-pointed Prym-Brill-Noether loci with special vanishing at two points, and determine their K-theory classes when the dimensions are as expected. The classes are derived by the applications of a formula for the K-theory of certain vexillary degeneracy loci in type D. In particular, we show a two-pointed version of Prym-Petri theorem on the expected dimension in the general case, with a coupled Prym-Petri map. Our approach is inspired by the work on pointed cases by Tarasca, and we generalize unpointed cases by De Concini-Pragacz and Welters.

Two-pointed Prym-Brill-Noether Loci and coupled Prym-Petri theorem

Abstract

We establish two-pointed Prym-Brill-Noether loci with special vanishing at two points, and determine their K-theory classes when the dimensions are as expected. The classes are derived by the applications of a formula for the K-theory of certain vexillary degeneracy loci in type D. In particular, we show a two-pointed version of Prym-Petri theorem on the expected dimension in the general case, with a coupled Prym-Petri map. Our approach is inspired by the work on pointed cases by Tarasca, and we generalize unpointed cases by De Concini-Pragacz and Welters.
Paper Structure (9 sections, 14 theorems, 89 equations, 1 figure)

This paper contains 9 sections, 14 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Degeneracy loci
  3. The two-pointed Prym-Brill-Noether loci
  4. The coupled Prym-Petri map
  5. General case
  6. On the quadruple
  7. The Kernel of the coupled Prym-Petri map
  8. A vanishing statement
  9. Proof of Main result

Key Result

Theorem 1.1

The dimension of $V_{\mathbf{a},\mathbf{b}}^r(C,\epsilon,P,Q)$ is at least $g-1-|\mathbf{a}+\mathbf{b}|$. If $V_{\mathbf{a},\mathbf{b}}^r(C,\epsilon,P,Q)$ has dimension of $g-1-|\mathbf{a}+\mathbf{b}|$, then it is Cohen-Macaulay, and in $CK^*(\mathscr{P}^\pm)[1/2]$.

Figures (1)

  • Figure 1:

Theorems & Definitions (21)

  • Theorem 1.1: = Theorem \ref{['thm:ck']}
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Theorem 3.1
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • Definition 4.3
  • ...and 11 more