Preheat grill to medium heat.
Cut four 14-x-14-inch squares of parchment paper. Fold each in half, and cut it into a wide heart shape. Then unfold and lay flat.
In a medium bowl, mix together potatoes with 1 tablespoon oil and a pinch of salt and pepper. Arrange a quarter of the potatoes on half of each of the parchment paper hearts, leaving a 1-inch border around the edge. Place a piece of fish on top of each. In the same bowl, combine the tomatoes, corn, green onions, garlic, basil, remaining 1 tablespoon oil, and salt and pepper.
Divide tomato and corn mixture evenly over fish, and drizzle a tablespoon of wine over each. Then fold over the empty half, and crimp the edges. Beginning at the top of the heart, roll the edges up and tightly over onto themselves, and twist the end so that the packet is securely sealed. Lay finished packets on a rimless cutting board, a pizza peel or on the back of a cookie sheet.
Using a wide spatula, carefully transfer sealed packets onto the grill, and cook for 10 minutes. (They should be puffed and golden brown.) Use the spatula to transfer the packets to individual plates. Let rest for 5 minutes, then slice open packets (or have diners do it themselves), and serve.