Rational divisors on Calabi–Yau threefolds

Question

Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are supported on the zero divisor $\mathbb{P}^1 \hookrightarrow \omega_{\mathbb{P}^1}$. The space $\omega_{\mathbb{P}^1}$ is the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}_2$ and the category $\mathcal{D}_0$ acts as a ‘local model’ for the derived category of a K3 surface near a ($-2$)-curve (with respect to a fixed $A_2$-configuration of spherical objects).

In [1], the authors consider one dimension higher: $\omega_{\mathbb{P}^2}$ is a quasi-projective Calabi–Yau threefold fibered over a projective plane, and one can analogously define a full subcategory of $D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^2})$ giving rise to quivery hearts related by a braid group action. My question is more or less how this is considered as a local model for projective Calabi–Yau threefolds: while CY threefolds contain rational curves, it’s not clear to me when they contain rational divisors.

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[1] Arend Bayer and Emanuele Macrì, The Space of Stability Conditions on the Local Projective Plane, Duke Math. J. 160 (2011), no. 2, 263–322, DOI 10.1215/00127094-1444249

[2] Tom Bridgeland, T-structures on some local Calabi–Yau varieties, Journal of Algebra 289 (2005), no. 2, 453-483, DOI https://doi.org/10.1016/j.jalgebra.2005.03.016.

Answer 1

Some CY threefolds do contain rational divisors. For instance, consider a general quintic $X \subset \mathbb{P}^4$ containing a plane $Z := \mathbb{P}^2 \subset \mathbb{P}^4$. Of course, $X$ is singular, but (by generality) its singularities are ordinary double points and they lie on $Z$. So, if you blowup $Z$, this will give you a small resolution of singularities $X' \to X$, so that $X'$ is a smooth CY threefold, and the strict transform of $Z$ is a rational surface in $X'$.

Answer 2

If you specifically wish for a projective plane to be contained in a projective CY3 $X$, start with a CY3 $Y$ with a $1/3(1,1,1)$ quotient singularity, and blow up the singular point to get the crepant resolution $f:X\to Y$ with exceptional locus $E\subset X$ isomorphic to the projective plane, with normal bundle being its canonical bundle. There are plenty of such $Y$ among hypersurfaces in weighted projective 4-space and toric fourfolds.