For K3 surfaces, every primitive base point free g^r_d with d≥4, r≥sqrt(d/2), and g>2d-3+(r-1)^2 has a Donagi-Morrison lift N adapted to |C| with Cliff(N⊗O_C)≤Cliff(A).
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A remark on the conjecture of Donagi-Morrison
For K3 surfaces, every primitive base point free g^r_d with d≥4, r≥sqrt(d/2), and g>2d-3+(r-1)^2 has a Donagi-Morrison lift N adapted to |C| with Cliff(N⊗O_C)≤Cliff(A).