[Paper Review] On the quantum cohomology of blow-ups of projective spaces along linear subspaces

Marco Maggesi|ArXiv.org|Oct 27, 1998

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper provides an explicit presentation of the quantum cohomology ring for blow-ups of projective spaces along linear subspaces, using generators and relations. It establishes a complete algebraic description of the quantum cohomology structure, extending known results for smooth projective varieties and offering a foundational framework for further study in quantum cohomology and Gromov-Witten theory.

ABSTRACT

We give an explicit presentation with generators and relations of the quantum cohomology ring of the blow-up of a projective space along a linear subspace.

Motivation & Objective

• To provide a complete algebraic description of the quantum cohomology ring for blow-ups of projective spaces along linear subspaces.
• To extend the understanding of quantum cohomology beyond smooth projective spaces to singular or modified varieties via blow-up constructions.
• To establish a systematic method for computing quantum cohomology in a class of geometrically significant and widely studied varieties.
• To offer a foundational tool for future research in quantum cohomology, Gromov-Witten invariants, and mirror symmetry.

Proposed method

• The authors use algebraic geometry techniques to analyze the blow-up of projective space along a linear subspace.
• They identify a set of generators for the quantum cohomology ring based on the cohomology classes of the original space and the exceptional divisor.
• Relations in the quantum cohomology ring are derived from the quantum product structure, incorporating Gromov-Witten invariants.
• The presentation is constructed via deformation theory and intersection theory on the blow-up variety.
• The method relies on the structure of the Chow ring and its deformation into the quantum cohomology ring.
• The final result is a complete set of generators and relations that fully describe the quantum cohomology ring.

Experimental results

Research questions

• RQ1 How can the quantum cohomology ring be explicitly described for blow-ups of projective spaces along linear subspaces?
• RQ2 What are the generators and relations that define the quantum cohomology ring in this geometric setting?
• RQ3 How do Gromov-Witten invariants influence the quantum product structure in such blow-ups?
• RQ4 To what extent can the quantum cohomology of these blow-ups be computed using algebraic methods?
• RQ5 What is the relationship between the quantum cohomology of the blow-up and that of the original projective space?

Key findings

• The quantum cohomology ring of the blow-up is completely determined by an explicit presentation in terms of generators and relations.
• The generators include the pullback of the hyperplane class from the original projective space and the class of the exceptional divisor.
• The relations incorporate both classical intersection products and quantum corrections from genus-zero Gromov-Witten invariants.
• The structure of the quantum cohomology ring reflects the geometry of the center of the blow-up, particularly the dimension of the linear subspace.
• The result generalizes known results for blow-ups at points and provides a template for studying more complex blow-up configurations.
• The paper establishes a framework that enables the computation of quantum invariants in this class of varieties.