Combine water, sugar, cinnamon and ginger in a large saucepan. Set over high heat and stir until sugar dissolves. When boiling, reduce heat. Simmer, uncovered, for 10 minutes.
Meanwhile, peel peaches, cut them in half and remove pits. Cut halves in half again. Rub peaches all over with cut lemon.
After sugar-ginger mixture has simmered for 10 minutes, add half of peaches. Reduce heat slightly and simmer gently until just tender, from 4 to 6 minutes.
Then remove with a slotted spoon. Add remaining peaches and repeat cooking as directed above.
When all peaches have been poached, remove ginger from poaching liquid and discard. Bring liquid back to a boil. Boil gently, uncovered, over medium-high heat until reduced to a thick amber syrup, about 10 minutes. Stir often near end of cooking. Remove cinnamon sticks.
Arrange peaches in bowls. Pour warm syrup over fruit. Serve immediately or cover and refrigerate. Just before serving reheat. Serve with ice cream or whipped cream.
