Once a chargeback is raised, the matter is adjudicated upon by the card scheme (MasterCard/Visa/etc.) before a decision is reached.
Not really, there's a significant amount of back and forth before it reaches the Arbitration stage. I'd be very surprised if a low value dispute over a train ticket ever escalated that far.
and would appreciate some more details as to what happens next.
If this is a Mastercard:
1. The issuer reviews your form. They may decide that it's not worth the hassle and just refund you, or that your claim is invalid or missing details. For the latter they'll contact you and discuss the situation.
2. Collaboration phase. If the issuer decides it's worth disputing, the transaction is submitted to the collaboration phase. This sits outside of the clearing process and gives the Acquirer opportunity to resolve the dispute by issuing a refund for the transaction, avoiding a chargeback. They should return the ARN of the refund in clearing, if they decide to do this.
3. If the Acquirer does not respond in collaboration then the issuer can submit a First Chargeback. It's likely the form you have filled in will be submitted verbatim or in extracts, to set out the cardholder dispute. An appropriate Reason Code will be selected by the issuer depending on the nature of your dispute (you haven't told us that). A First Chargeback is submitted to clearing.
4. The Acquirer may accept the First Chargeback. If not, they will submit a Second Presentment citing an appropriate Reason Code and supplying evidence supporting that, if applicable. That may simply be the ARN in DE 72 to identify the refund previously processed.
Most chargebacks are resolved at First Chargeback or Second Presentment.
Beyond that we get into pre-arbitration and arbitration cases. These are expensive and labour intensive.