  The purpose of this paper is to prove that one can read off the gonality
sequence of a smooth projective curve from syzygies of secant varieties of the
curve embedded by a line bundle of sufficiently large degree. More precisely,
together with Ein-Niu-Park's theorem, our main result shows that the gonality
sequence of a smooth projective curve completely determines the shape of the
minimal free resolutions of secant varieties of the curve of sufficiently large
degree. This is a natural generalization of the gonality conjecture on syzygies
of smooth projective curves established by Ein-Lazarsfeld and Rathmann to the
secant varieties.
