  We introduce a multi-parameter deformation of the triply-graded
Khovanov--Rozansky homology of links colored by one-column Young diagrams,
generalizing the "$y$-ified" link homology of Gorsky--Hogancamp and work of
Cautis--Lauda--Sussan. For each link component, the natural set of deformation
parameters corresponds to interpolation coordinates on the Hilbert scheme of
the plane. We extend our deformed link homology theory to braids by introducing
a monoidal dg 2-category of curved complexes of type $A$ singular Soergel
bimodules. Using this framework, we promote to the curved setting the
categorical colored skein relation from arXiv:2107.08117 and also the notion of
splitting map for the colored full twists on two strands. As applications, we
compute the invariants of colored Hopf links in terms of ideals generated by
Haiman determinants and use these results to establish general link splitting
properties for our deformed, colored, triply-graded link homology. Informed by
this, we formulate several conjectures that have implications for the relation
between (colored) Khovanov--Rozansky homology and Hilbert schemes.
