[Paper Review] Solomon's Relations for Welshinger's Invariants: Examples

Xujia Chen|arXiv (Cornell University)|Sep 24, 2018

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper derives and applies WDVV-style relations for Welschinger's invariants in real symplectic fourfolds, specializing them to the complex projective plane, its real blowups, and the quadric surface under two involutions. Using these relations, it computes low-degree Welschinger invariants via Mathematica and provides extensive tables of numerical invariants, offering a systematic computational framework for real enumerative geometry in these spaces.

ABSTRACT

For the convenience of use, this note presents the WDVV-style relations for Welschinger's invariants counting real curves in real symplectic fourfolds, announced by Solomon in 2007 and recently established by the author, specializes the two relations to the projective plane, its real blowups, and the quadric surface with two different involutions, and includes tables of some low-degree invariants of these spaces obtained from these relations with Mathematica. It also includes extensive tables of Welschinger's invariants in low degrees.

Motivation & Objective

• To present and apply Solomon's WDVV-style relations for Welschinger's invariants in real symplectic fourfolds.
• To specialize these relations to concrete real symplectic fourfolds: the complex projective plane, its real blowups, and the quadric surface with two distinct involutions.
• To compute and tabulate low-degree Welschinger invariants for these spaces using symbolic computation via Mathematica.
• To provide a comprehensive reference of numerical invariants for researchers in real enumerative geometry.

Proposed method

• Derives WDVV-style relations for Welschinger's invariants based on Solomon's 2007 announcement and subsequent proof.
• Applies the general relations to specific real symplectic fourfolds: CP², its real blowups, and the quadric surface with two involutions.
• Uses the specialized relations as recursive constraints to compute invariants in low degrees.
• Employs Mathematica to perform symbolic computations and generate extensive tables of invariants.
• Validates consistency of results by checking integrality and sign patterns expected from real algebraic geometry.
• Presents the invariants in structured tables for immediate use in further research.

Experimental results

Research questions

• RQ1 How do Solomon's WDVV-style relations for Welschinger invariants specialize to the complex projective plane and its real blowups?
• RQ2 What are the low-degree Welschinger invariants for the quadric surface under the two different real structures (involutions)?
• RQ3 Can the WDVV relations be effectively used to compute and cross-verify Welschinger invariants in real symplectic fourfolds?
• RQ4 What patterns or structural properties emerge in the computed tables of invariants across different real symplectic fourfolds?
• RQ5 How do the invariants behave under blowup operations in the real setting, and can this be captured via the WDVV relations?

Key findings

• The WDVV-style relations successfully constrain and compute Welschinger invariants in low degrees for CP² and its real blowups.
• Explicit tables of Welschinger invariants are generated for the complex projective plane up to low degrees, providing a reference for future work.
• For the quadric surface with two distinct involutions, the relations yield distinct sets of invariants, reflecting the different real structures.
• The computed invariants are integral and exhibit consistent sign patterns, supporting their geometric interpretation.
• The use of Mathematica enables efficient and reliable computation of invariants that would be infeasible by hand.
• The tables of invariants are extensive and systematically organized, offering immediate utility for researchers in real enumerative geometry.