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We present a novel technique to obtain base independent expressions for thematter loci of fibrations of complete intersection Calabi-Yau onefolds in toricambient spaces. These can be used to systematically construct elliptically andgenus one fibered Calabi-Yau $d$-folds that lead to desired gauge groups andspectra in F-theory. The technique, which we refer to as GV-spectroscopy, isbased on the calculation of fiber Gopakumar-Vafa invariants using theBatyrev-Borisov construction of mirror pairs and application of the so-calledFrobenius method to the data of a parametrized auxiliary polytope. Inparticular for fibers that generically lead to multiple sections, onlymulti-sections or that are complete intersections in higher codimension, ourtechnique is vastly more efficient than classical approaches. As an applicationwe study two Higgs chains of six-dimensional supergravities that are engineeredby fibrations of codimension two complete intersection fibers. Both chains endon a vacuum with $G=\mathbb{Z}_4$ that is engineered by fibrations ofbi-quadrics in $\mathbb{P}^3$. We use the detailed knowledge of the structureof the reducible fibers that we obtain from GV-spectroscopy to comment on thecorresponding Tate-Shafarevich group. We also show that for all fibers thesix-dimensional supergravity anomalies including the discrete anomaliesgenerically cancel.

GV-Spectroscopy for F-theory on genus-one fibrations