Canonical bundle of a Weierstrass hypersurface

On a trivializing analytic open set for $L$, the weighted projective bundle is the quotient of ${\mathbb{C}}^3\setminus \{0\}$ by the action $\lambda \cdot (x,y,z)=({\lambda }^{2}x,{\lambda }^{3}y,\lambda z)$. On $W^{\circ }$ the vertical Euler sequence is $0⟶{\mathcal{O}}_{W^{\circ }}⟶({\mathcal{O}}_W(2)\otimes p^{\ast }{L}^2)\oplus ({\mathcal{O}}_W(3)\otimes p^{\ast }{L}^3)\oplus {\mathcal{O}}_W(1)⟶T_{W^{\circ }\mathrm{/}{B}^{\mathrm{a}\mathrm{n}}}⟶0.$ Indeed the first map is tangent to the scaling orbit, with local vector $2x{\mathrm{\partial }}_x+3y{\mathrm{\partial }}_y+z{\mathrm{\partial }}_z$, and the quotient by that orbit is the vertical tangent bundle; the transitions in Weierstrass model over a line bundle give the displayed line-bundle factors. Taking determinants gives $\det T_{W^{\circ }\mathrm{/}{B}^{\mathrm{a}\mathrm{n}}}\cong {\mathcal{O}}_W(6)\otimes p^{\ast }{L}^{\otimes 5}$, hence ${\omega }_{W^{\circ }\mathrm{/}{B}^{\mathrm{a}\mathrm{n}}}\cong {\mathcal{O}}_W(-6)\otimes p^{\ast }((L^{\vee }{)}^{\otimes 5})$.

If $z\mathrm{\ne }0$, the weight-one coordinate gives a smooth analytic chart. If $z=0$ on $Y$, then $y^2=x^3$; neither $x$ nor $y$ can vanish, so the stabilizer is ${\mu }_2\cap {\mu }_3=\{1\}$. Thus $Y\subset W^{\circ }$. The defining section is not identically zero on any component, since the local homogeneous polynomial has distinct $y^2$ and $x^3$ terms with coefficient $1$. The local rings of the smooth analytic ambient $W^{\circ }$ are domains, so this section is a nonzerodivisor along $Y$ and cuts out a Cartier hypersurface.

Let $M={\mathcal{O}}_W(6)\otimes p^{\ast }{L}^{\otimes 6}$, the holomorphic line bundle of the defining section. In a local analytic trivialization of $M$ and ${\omega }_{W^{\circ }}$, write the equation as a nonzerodivisor $f$ in the regular analytic local ring $A$. The resolution $0\to A\stackrel{f}{\to }A\to A\mathrm{/}(f)\to 0$ gives ${\mathrm{Ext}}_A^1(A\mathrm{/}(f),A)=A\mathrm{/}(f)$; restoring the line-bundle trivializations identifies the analytic lci dualizing line with $({\omega }_{W^{\circ }}\otimes M){\mathrm{\mid }}_Y$. The ambient canonical line on $W^{\circ }$ is ${\omega }_{W^{\circ }}={\omega }_{W^{\circ }\mathrm{/}{B}^{\mathrm{a}\mathrm{n}}}\otimes p^{\ast }{K}_{B^{\mathrm{a}\mathrm{n}}}$. Substituting step 1.1 and $M$ gives ${\mathcal{O}}_W(-6)\otimes p^{\ast }((L^{\vee }{)}^{\otimes 5})\otimes p^{\ast }{K}_{B^{\mathrm{a}\mathrm{n}}}\otimes {\mathcal{O}}_W(6)\otimes p^{\ast }{L}^{\otimes 6}$, which restricts to $p_Y^{\ast }(K_{B^{\mathrm{a}\mathrm{n}}}\otimes L)$. Hence $L\cong K_{B^{\mathrm{a}\mathrm{n}}}^{-1}$ makes the analytic dualizing sheaf trivial.

The broad convention in [F2] requires smoothness, connectedness, compactness, a Kähler form, and a holomorphic trivialization of the canonical line bundle. Step 2.1 supplies only the last line-bundle condition when $L\cong K_{B^{\mathrm{a}\mathrm{n}}}^{-1}$; therefore the result does not by itself identify a singular Weierstrass hypersurface with a Calabi–Yau manifold in that convention.