Crepant resolution and matter surface

Let $B$ be a smooth integral complex algebraic threefold and let $\mathcal{L}\to B$ be an algebraic line bundle. Let $p:W\to B$ be the weighted projective bundle of ${\mathcal{L}}^{\otimes 2}\oplus {\mathcal{L}}^{\otimes 3}\oplus {\mathcal{O}}_B$ with coordinate weights $(2,3,1)$. In local trivializations the coordinate twists are $x\in {\mathcal{O}}_W(2)\otimes p^{\ast }{\mathcal{L}}^{\otimes 2}$, $y\in {\mathcal{O}}_W(3)\otimes p^{\ast }{\mathcal{L}}^{\otimes 3}$, and $z\in {\mathcal{O}}_W(1)$. Let $Y_W\subseteq W$ be the integral hypersurface $y^2=x^3+fx{z}^4+g{z}^6$, where $f$ and $g$ are algebraic sections of ${\mathcal{L}}^{\otimes 4}$ and ${\mathcal{L}}^{\otimes 6}$, respectively, and set ${\pi }_W=p{\mathrm{\mid }}_{Y_W}$. Let $W^{\circ }\subseteq W$ be its smooth open containing $Y_W$. Write $i:Y_W\hookrightarrow W^{\circ }$ for the hypersurface inclusion and $M={\mathcal{O}}_W(6)\otimes p^{\ast }{\mathcal{L}}^{\otimes 6}{\mathrm{\mid }}_{W^{\circ }}$ for the line bundle of its defining section. The integral fourfold hypersurface is the zero subscheme of a nonzero section of $M$, hence an effective Cartier divisor in the smooth $W^{\circ }$. Define its dualizing (canonical) sheaf by the local complete-intersection Ext formula ${\omega }_{Y_W}:=\mathcal{E}x{t}_{{\mathcal{O}}_{W^{\circ }}}^1\text{\hspace{0.17em}⁣}(i_{\ast }{\mathcal{O}}_{Y_W},\det {\mathrm{\Omega }}_{W^{\circ }\mathrm{/}\mathbb{C}}^1),$ regarded as an ${\mathcal{O}}_{Y_W}$-module via the closed immersion $i$. The Cartier resolution $0\to M^{-1}\to {\mathcal{O}}_{W^{\circ }}\to i_{\ast }{\mathcal{O}}_{Y_W}\to 0$ computes this as the invertible line $i^{\ast }(\det {\mathrm{\Omega }}_{W^{\circ }\mathrm{/}\mathbb{C}}^1\otimes M)$. For a smooth complex algebraic fourfold $\widehat{Y}$, set ${\omega }_{\widehat{Y}}=\det {\mathrm{\Omega }}_{\widehat{Y}\mathrm{/}\mathbb{C}}^1$.

A crepant resolution of $Y_W$ is specified data $\rho :\widehat{Y}⟶Y_W$ where $\widehat{Y}$ is a smooth integral complex algebraic fourfold, $\rho$ is proper (separated, of finite type, and universally closed) and birational (it maps the generic point to the generic point and induces an isomorphism of function fields), and ${\omega }_{\widehat{Y}}\cong {\rho }^{\ast }{\omega }_{Y_W}$ as line bundles. In this carrier, “crepant” means exactly this canonical-line isomorphism. Set $\widehat{\pi }:={\pi }_W\circ \rho$. For $b\in B$, a geometric fiber means base change along $\mathrm{Spec}\overline{\kappa (b)}\to B$. A resolution chamber is the specified resolution data $(\rho ,\mathcal{C})$, where $\mathcal{C}$ is a specified collection of irreducible curve components in geometric fibers of $\widehat{\pi }$, including the chosen components used to form matter surfaces below. Matter surfaces attached to different resolution chambers are separate data. This definition does not assert that a resolution exists, is unique, projective, or flat over $B$.

If $T\subseteq B$ is an integral closed subvariety with generic point ${\eta }_T$, its geometric generic fiber is the scheme ${\widehat{Y}}_{{\bar{\eta }}_T}:=\widehat{Y}{\times }_B\mathrm{Spec}\overline{\kappa ({\eta }_T)}$, where $\overline{\kappa ({\eta }_T)}$ is an algebraic closure of the residue field at ${\eta }_T$. Its fiber configuration means the isomorphism class of this scheme. A codimension-two fiber-change locus relative to a specified integral closed divisor $D\subset B$ is a specified collection of irreducible reduced closed curves $\mathrm{\Sigma }\subset D$ such that ${\dim }_{\mathbb{C}}\mathrm{\Sigma }=1$ (hence ${\mathrm{codim}}_B\mathrm{\Sigma }=2$) and the geometric generic fibers over ${\eta }_{\mathrm{\Sigma }}$ and ${\eta }_D$ are nonisomorphic as schemes after base change along any chosen pair of $\mathbb{C}$-embeddings of their residue-field algebraic closures into a common algebraically closed field. A matter curve is one such specified ${\mathrm{\Sigma }}_R$; $D$ is typically a chosen discriminant component.

A selected fibral curve family is a specified closed reduced irreducible surface ${\mathcal{S}}_R\hookrightarrow \widehat{Y}$ with $\widehat{\pi }({\mathcal{S}}_R)\subseteq {\mathrm{\Sigma }}_R$ such that the restriction $q_R:{\mathcal{S}}_R\to {\mathrm{\Sigma }}_R$ is proper (separated, of finite type, and universally closed) and flat (each ${\mathcal{O}}_{{\mathcal{S}}_R,s}$ is a flat ${\mathcal{O}}_{{\mathrm{\Sigma }}_R,q_R(s)}$-module), every geometric fiber is nonempty and pure of dimension one, and its geometric generic fiber is isomorphic to ${\mathbb{P}}^1$ and is one of the chosen components in $\mathcal{C}$. The surface ${\mathcal{S}}_R$ is the matter surface for that resolution chamber. Its complex dimension is two; its associated integral algebraic 2-cycle $[{\mathcal{S}}_R]$ is the coefficient-one generator of the free abelian group on irreducible two-dimensional closed subvarieties of $\widehat{Y}$. The definition provides no identification between data attached to distinct choices.

Birational morphisms of integral finite-type schemes

Flat morphism of schemes

Weierstrass model over a line bundle

Proper morphisms

Weigand, §§4.5–5.6

The square commutes because $q_R$ is the restriction of $\widehat{\pi }$ to ${\mathcal{S}}_R$; it records the surface family and base curve used in the definition.

The matter surface is a geometric cycle in a chosen resolution. It is not, by definition, a representation, a gauge algebra, a chiral index, or evidence that a crepant resolution exists.