  It is well known that the Eisenbud-Goto regularity conjecture is true for
arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces,
smooth threefolds in $\mathbb{P}^5$, and toric varieties of codimension two.
After J. McCullough and I. Peeva constructed counterexamples in 2018, it has
been an interesting question to find the categories such that the Eisenbud-Goto
conjecture holds. So far, surface counterexamples have not been found while
counterexamples of any dimension greater or equal to 3 are known.
  In this paper, we construct counterexamples to the Eisenbud-Goto conjecture
for projective surfaces in $\mathbb{P}^4$ and investigate projective
invariants, cohomological properties, and geometric properties. The
counterexamples are constructed via binomial rational maps between projective
spaces.
