Syzygies of tangent-developable surfaces and K3 carpets via secant varieties

We give simple geometric proofs of Aprodu, Farkas, Papadima, Raicu and Weyman’s theorem on syzygies of tangent-developable surfaces of rational normal curves and Raicu and Sam’s result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green’s conjecture for general curves of genus $g$ over an algebraically closed field $k<!--FUNCTION\; APPLICATION-->$ with $\mathrm{char}<!--FUNCTION\; APPLICATION-->(k<!--FUNCTION\; APPLICATION-->)=0$ or $\mathrm{char}<!--FUNCTION\; APPLICATION-->(k<!--FUNCTION\; APPLICATION-->)\ge \lfloor (g-1)∕2\rfloor$. Our approach provides a new way to study tangent-developable surfaces in general. Along the way, we show the arithmetic normality of tangent-developable surfaces of arbitrary smooth projective curves of large degree.