  We show that a certain two-dimensional family of Weyl sums of length $P$
takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the
unit torus, contradicting a widely held expectation that Weyl sums should
exhibit square-root cancellation on generic subvarieties of the unit torus.
This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G.
Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general
polynomials of arbitrary degree. The new ingredients of our approach are the
classical results of E. Bombieri (1966) on exponential sums along a curve and
R. J. Duffin and A. C. Schaeffer (1941) on Diophantine approximations by
rational numbers with prime denominators.
