Mirror symmetry and projective geometry of Reye congruences I

TL;DR

This work addresses mirror symmetry for the Reye congruence Calabi–Yau $X$ in $\mathbb{P}^4$, identifying a nontrivial Fourier–Mukai partner $Y$ as a double cover of a Hessian/determinantal quintic. The authors construct the mirror family ${\mathcal{X}}^\vee$ via the Batyrev–Borisov framework and analyze its GKZ Picard–Fuchs system, revealing two large complex structure limits and predicting a second Calabi–Yau $Y$ with invariants $(\deg(Y), c_2\cdot H, e, h^{1,1}, h^{2,1})=(10,40,-50,1,26)$; they prove the existence of $Y$ through a geometric construction involving the Hessian quintic $H$ and a ramified double cover, and conjecture a derived-equivalence $D(Coh(X))\cong D(Coh(Y))$, linking to homological projective duality. Finally, they compute genus-zero and higher-genus Gromov–Witten/BPS numbers for $X$ and $Y$ by mirror symmetry, providing explicit curve counts with tables in the appendices.

Abstract

Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mP^4$ branched over a curve. We also calculate BPS numbers of both $X$ and $Y$ (and also a related Calabi-Yau complete intersection $\tilde X_0$) using mirror symmetry.

Theorems & Definitions (42)

• Proposition 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Remark 2.5
• Proposition 2.6
• proof