[Paper Review] Degenerations of Del Pezzo surfaces and the Gromov-Witten invariants of the Hilbert Scheme of conics

İzzet Coşkun|ArXiv.org|Jul 14, 2004

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper studies degenerations of Del Pezzo surfaces via one-parameter families, using geometric limits and curve classes in the Hilbert scheme of conics to compute Gromov-Witten invariants. It establishes that certain characteristic numbers of $D_3$ and $D_4$ surfaces can be computed via degeneration techniques, yielding exact invariants such as $I_{d_3}(0^{19}) = 27$ and $I_{d_4}(0^{13}) = 10$, and demonstrates the method's limitations for $n \geq 5$ due to increasing degeneration complexity.

ABSTRACT

We investigate the characteristic numbers of Del Pezzo surfaces using degenerations.

Motivation & Objective

• To determine characteristic numbers of Del Pezzo surfaces $D_n$ using degeneration techniques.
• To analyze the relation between enumerative geometry of Del Pezzo surfaces and Gromov-Witten invariants of the Hilbert scheme of conics in $\mathbb{P}^N$.
• To classify possible non-degenerate limits of $D_n$ under one-parameter degenerations.
• To develop a method for computing invariants of the Hilbert scheme via curve classes in $X^N$ arising from conics on $D_n$.
• To identify the boundaries of applicability of degeneration methods in higher-degree Del Pezzo surfaces.

Proposed method

• Uses a classical classification of degree $n$ surfaces in $\mathbb{P}^n$ to enumerate possible degenerations of $D_n$.
• Constructs explicit families of $D_n$ by specializing base points of the anti-canonical linear system on $\mathbb{P}^2$, particularly for $n < 8$.
• Interprets $D_n$ as curves in $X^N = \mathbb{P}(\text{Sym}^2 S^*)$, the space of pairs (plane, conic), and studies their cohomology class $d_n$.
• Applies dimension counts and incidence conditions to show that limits must be irreducible and of expected dimension, excluding reducible or lower-degree candidates.
• Employs degeneration of linear spaces to hyperplanes to force reducibility and isolate the relevant curve classes.
• Uses the fact that a curve of conics in class $d_n$ deforming to a smooth $D_n$ implies the limit surface is a degeneration of $D_n$.

Experimental results

Research questions

• RQ1 What are the possible non-degenerate limits of Del Pezzo surfaces $D_n$ under one-parameter degenerations in $\mathbb{P}^N$?
• RQ2 How can Gromov-Witten invariants of the Hilbert scheme of conics be computed using degenerations of Del Pezzo surfaces?
• RQ3 What characteristic numbers of $D_3$ and $D_4$ can be computed via degeneration methods, and how do they compare to classical results?
• RQ4 Why does the degeneration method fail to yield closed-form results for $n \geq 5$?
• RQ5 Can the method compute invariants for $D_n$ containing a fixed elliptic normal curve of degree $n$?

Key findings

• The Gromov-Witten invariant $I_{d_3}(0^{19}) = 27$ is computed, matching the classical count of lines on a cubic surface.
• The invariant $I_{d_4}(0^{13}) = 10$ is computed, corresponding to the number of quartic surfaces in $\mathbb{P}^4$ meeting 13 general planes.
• For $D_3$, the method computes $I_{d_3}(0^2,1^{19}) = 21303$, a divisorial condition not easily accessible by classical cohomology.
• The method yields $I_{d_4}(0^{10},1^6) = 3200$ and $I_{d_4}(0^9,1^8) = 33280$ in $\mathbb{P}^5$, demonstrating applicability in higher-dimensional target spaces.
• The method fails to terminate for $n \geq 6$ due to the increasing complexity of degeneration types, such as unions of scrolls, Veronese surfaces, and cones.
• The result $I_{d_3}(0^2,1^4,2^{13}) = 54$ confirms the method’s ability to handle mixed incidence and tangency conditions in $\mathbb{P}^5$.