Shioda projection and torsional coweight data

Let $B$ be a smooth projective complex threefold and let $\pi :\widehat{Y}\to B$ be a smooth projective fourfold with a specified elliptic-fibration-with-section structure as in Elliptic fibration with a section. Assume that $\widehat{Y}$ is a chosen smooth resolution of a Weierstrass model as in Crepant resolution and matter surface; existence, projectivity, and elliptic flatness are hypotheses on this particular input. Let $O$ be the zero-section and let $S$ be another specified rational section, viewed by closure as a divisor class. Fix a smooth gauge divisor $D\subset B$. The root datum used here is specified by an integer matrix $A=(A_{ij}{)}_{1\le i,j\le r}$, where $r\ge 0$, $A_{ii}=2$, $A_{ij}\in \{0,-1\}$ for $i\mathrm{\ne }j$, and $A$ is symmetric positive definite when $r>0$. These conditions define finite-type simply-laced Cartan data; no other root-system structure is used below. Let $V$ and $V^{\vee }$ be real vector spaces with bases ${\alpha }_1,\dots ,{\alpha }_r$ and ${\alpha }_1^{\vee },\dots ,{\alpha }_r^{\vee }$, respectively, and define their bilinear root-coroot pairing by $⟨{\alpha }_i,{\alpha }_j^{\vee }⟩=A_{ij}$. The simple roots and coroots are these basis vectors. Define the root and coroot lattices by $Q={\sum }_i\mathbb{Z}{\alpha }_i$ and $Q^{\vee }={\sum }_i\mathbb{Z}{\alpha }_i^{\vee }$. If $r=0$, take $V=V^{\vee }=0$, $A$ to be the unique endomorphism of the zero space, and all basis-indexed sums to be empty.

Assume the chosen resolution has Cartan divisors $E_1,\dots ,E_r$ and specified integral fibral curves $C_1,\dots ,C_r$ over the generic point of $D$, with $E_i\cdot C_j=-A_{ij}$. Intersections are degrees of the restricted divisor line bundles; all divisor classes and push-forwards below are taken in rational numerical equivalence, with proper push-forward along $\pi$. Define the Shioda projection datum of $S$ by

For a finite-order section $T$ in the Mordell–Weil group of the generic fiber, put ${\delta }_j(T)=([T]-[O])\cdot C_j$ and define

The coweight lattice is $P^{\vee }=\{\lambda \in V^{\vee }:⟨{\alpha }_j,\lambda ⟩\in \mathbb{Z}\text{ for every }j\}$; the associated fractional coweight datum is the class of ${\lambda }_T$ in $P^{\vee }\mathrm{/}{Q}^{\vee }$. For $r=0$, both $P^{\vee }$ and $Q^{\vee }$ are $\{0\}$. No claim that this class is nonzero, that the Shioda assignment is a group homomorphism, or that a physical gauge group is determined is part of this definition.

Elliptic fibration with a section

Crepant resolution and matter surface

The Axiom of Choice

T. Weigand, TASI Lectures on F-theory, §7.1 and §7.7

Morrison–Park, §§1 and 3.1

This item supplies only a local algebraic definition and the two displayed matrix checks. It does not prove that a given fibration has a projective crepant resolution, that the Mordell–Weil group is finitely generated, that torsion is nonzero, or that the coweight class acts as a gauge-group quotient. Those are separate geometric or physical inputs.