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I'm hoping that my visa will just change to a normal Schengen, tourist visa and I won't need permesso di soggiorno.
The day after your 3 month student visa (long-stay/national visa) expires will be treated as the first day of the 90/180 day period for visa free nationals.
Note: the present Schengen Border Code does not state this explicidly, but an amendment of the Schengen Border Code, that will come into force when the Entry/Exit System is introduced, will in Article 12(3) quoted below.
Some countries have, in the past, required persons to leave the Schengen Area just to get an entry stamp when returning.
The present "Practical Handbook for Border Guards (Schengen Handbook)" states that this is not compatible with the Schengen rules.
National rules or practices which lay down that a third-country national must as a general rule leave its territory after termination of a long-stay (covered by a residence permit or national visa) before being allowed to start a subsequent short-term stay on that territory are not compatible with Schengen rules.
Article 12
(Presumption as regards fulfilment of conditions of duration of short stay)
...
3. The presumption referred to in paragraph 1 may be rebutted where the third-country national provides, by any means, credible evidence, such as transport tickets or proof of presence outside the territory of the Member States or of the date of expiry of a previous residence permit or long-stay visa, that he or she has respected the conditions relating to the duration of a short stay.
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