  Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d
F-theory compactifications, we combine machine learning and algebraic geometry
techniques to analyze line bundle cohomologies on families of holomorphic
curves. To quantify jumps of these cohomologies, we first generate 1.8 million
pairs of line bundles and curves embedded in $dP_3$, for which we compute the
cohomologies. A white-box machine learning approach trained on this data
provides intuition for jumps due to curve splittings, which we use to construct
additional vector-like Higgs-pairs in an F-Theory toy model. We also find that,
in order to explain quantitatively the full dataset, further tools from
algebraic geometry, in particular Brill--Noether theory, are required. Using
these ingredients, we introduce a diagrammatic way to express cohomology jumps
across the parameter space of each family of matter curves, which reflects a
stratification of the F-theory complex structure moduli space in terms of the
vector-like spectrum. Furthermore, these insights provide an algorithmically
efficient way to estimate the possible cohomology dimensions across the entire
parameter space.
