What is the dietary deficiency that causes dryness and fine red lines in the palms of the hands?

Depends: Dry skin and redness is most often caused by lifestyle behaviors, such as frequent hand washing or exposure to harsh chemicals without proper moisturization. Some medical conditions, such as thyroid disorders or diabetes may increase your susceptibility to dryness. There is a protein called filaggrin that is altered in some people with eczema or a severe form of dry skin, called ichthyosis.

There is none: There is no evidence that i'm aware of that any dietary deficiency causes dryness of the hands. Most hand dermatitis is environmental, from exposure to chemicals, harsh soaps, cold/dry air, etc. Effective treatments are available, but you'll have to consult a dermatologist.