Assunta is the Italian form of Spanish Asunción, which basically means, "Assumption." It is derived from the Latin "Absunta," and is a traditional and very common name in means that it reflects the devotion for the Virgin Mary.
The name Assunta comes from the Assumption (of the Blessed Virgin), not the Ascension (of Jesus Christ). Most of the time a girl is named Assunta Maria or Maria Assunta.