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Gromov-Witten and Welschinger invariants of del Pezzo varieties

Thi-Ngoc-Anh Nguyen

Abstract

In this paper, we establish formulas for computing genus-$0$ Gromov-Witten and Welschinger invariants of some del Pezzo varieties of dimension three by comparing to that of dimension two. These formulas are generalizations of that given in three-dimensional projective space by E. Brugallé and P. Georgieva in 2016.

Gromov-Witten and Welschinger invariants of del Pezzo varieties

Abstract

In this paper, we establish formulas for computing genus- Gromov-Witten and Welschinger invariants of some del Pezzo varieties of dimension three by comparing to that of dimension two. These formulas are generalizations of that given in three-dimensional projective space by E. Brugallé and P. Georgieva in 2016.
Paper Structure (41 sections, 56 theorems, 185 equations, 7 figures, 5 tables)

This paper contains 41 sections, 56 theorems, 185 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction and statement of the main results
  2. Introduction and motivation
  3. Main results
  4. Pencils of surfaces in the linear system |-(1/2)K_ X|
  5. Properties of pencils of surfaces in the linear system $\boldsymbol{|-\frac{1}{2}K_X|}$
  6. Monodromy
  7. Enumeration on pencils of surfaces
  8. Real pencils of real del Pezzo surfaces
  9. Properties of real pencils of real surfaces
  10. Enumerations on real pencils of real del Pezzo surfaces
  11. Rational curves on del Pezzo varieties of dimension 3 and 2: Relationship
  12. Specifications of configuration of points
  13. Moduli spaces and enumeration of balanced rational curves
  14. Moduli spaces of stable maps and balanced rational curves
  15. Enumeration of balanced rational curves
  16. ...and 26 more sections

Key Result

Theorem 1.3

Let $X$ denote a $3$-dimensional del Pezzo variety such that $\ker(\psi) \cong \mathbb{Z}$, where $\Sigma$ and $\psi$ are as described above. Let $S$ be a generator of $\ker (\psi)$. For every homology class $d\in H_2(X;\mathbb{Z})$, one has

Figures (7)

  • Figure 1: Example of a monodromy transformation where the general fibers $\Sigma$ are quadric surfaces and the singular ones are quadric cones in $\mathbb{C}P^3$.
  • Figure 2: Real divisors in the Picard group $\mathbb{R}(\mathop{\mathrm{Pic}}\nolimits(E)/_{x \sim \overline{D}_{\phi}-x})$ of a real elliptic curve with non-empty real part.
  • Figure 3: The real solutions of Equation (\ref{['eq-solutions.in.PicE']}) corresponding to the three cases of Corollary \ref{['coro-degree.case.real']}.
  • Figure 4: A direct orthogonal frame $(v_1(x_0),v_2(x_0)),v_3(x_0)$ at $x_0=f(\xi)\in \mathbb{R} C$ of the vector bundle $T \mathbb{R} X|_{\mathbb{R} C}$.
  • Figure 5: $L_0$ and $T(L_0)$ rotate in opposite directions in $\mathbb{R} X$.
  • ...and 2 more figures

Theorems & Definitions (138)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 128 more