This varies a bit in different versions of the story. Half with Hades and half with her mother Demeter is the most common .
Persephone once lived with her mother, Demeter. However, she was abducted by Hades. After an altercation between Demeter and Hades, it was decided that Persephone would resides with Hades for 3 months of the year. The other months of the year she lives with her mother.
It is likely that Persephone dwelt with Demeter either upon Olympus or the earth.
It depends on what time of the year these months are during. Winter and Autumn are associated with the Underworld, while Spring and Summer are associated with Persephone returned to Earth.
She spends 6 months per year in the Underworld.
Persephone was the daughter of Demeter, the goddess of plants. When Persephone was kidnapped by Hades, she stopped taking care of plants and the plants died. This became winter. Persephone was later returned but she had eaten some seeds. So she lived half of the year with her mother and the plants flourished. The other half of the year, she had to go back to Hades and Demeter would again stop caring for the plants.
Persephone did return to spend half the year on Earth between her mother, Demeter, and then to the Underworld to be with her husband, Hades.
In Greek myth, Persephone never died, by eating of the food of the Underworld she for half the year she had to spend below with her husband Hades, and half above with her mother Demeter.
Demeter (mother of Persephone who spend half the year in Hades with her husband, giving us the seasons)
Persephone only spends 6 months in Hades. -Indiriel ________________ In the underworld
Persephoen has one third of the year is spent in the Underworld with Hades, while two-thirds of it is spent with her mother Demeter.
The Underworld with Hades, her husband.
Persephone was not punished. She ate the food of the Underworld, so had to spend part of a year therein with her husband Hades.
Persephone spends Spring/Summer/Autumn with Demeter, her mother.
Demeter had to compromise with allowing Persephone to spend part of the year in the Underworld with Hades because of the agreement struck between Hades and Zeus. This compromise was the result of Persephone eating six pomegranate seeds in the Underworld, which bound her to spend a portion of each year there.
She had to be away from her mother, Demeter, for half the year.
6 months
Autumn-Winter