Sorry Trish, I've been miss-reading your question from the start! Apologies.
Yes, the Sigma 10-20 DC will fit and function on your camera in DX mode, but you won't be using the full potential of the D610's full-frame sensor, and more than half the area will be wasted. You should be looking at FX lenses, DG in Sigma's parlance, or Di for Tamron. 16-35-ish kind of range.
A good general purpose focal length range is 24-70 (Nikon, Sigma, Tamron), or 24-120 (Nikon) or something similar, depending on budget. My choice would be the Tamron 24-70/2.8 VC.
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