To make the crust: Pulse the pecans, flour and sugar in a food processor until the nuts are finely ground. Add the butter and pulse just until combined. Press the crust evenly over the bottom and 1 inch up the sides of the pan. Bake for 10 to 12 minutes, until light brown. Transfer to a wire rack and cool completely. Reduce the oven temperature to 300 F.

To make the filling: Mash the blackberries and sugar together in a medium saucepan with a pastry blender or a fork. Cook over medium heat, stirring occasionally, until the berries begin to release their juices. Stir in the cornstarch, bring to a boil over medium-high heat, stirring constantly, and boil for 1 minute. Pour the purée through a coarse strainer set over a small bowl, pressing hard on the solids to extract as much liquid as possible. Refrigerate, tightly covered, until thoroughly chilled.

With an electric mixer on medium-high speed, beat the cream cheese in a large bowl for about 2 minutes, until light and fluffy. Gradually beat in the condensed milk, zest, lime juice, and vanilla, scraping down the side of the bowl as necessary. Reduce the speed to medium and add the eggs one at a time, beating well after each addition. Pour the batter into the pan.

Transfer the blackberry purée to a small glass measure. Drizzle it in a spiral pattern over the batter, then swirl a table knife through the batter to marbleize it. Bake for 55 to 60 minutes, until the cheesecake is puffed on the sides and still slightly jiggly in the center. Let cool on a wire rack.

Refrigerate the cheesecake, tightly covered, for at least 8 hours, until thoroughly chilled and set, or for up to 2 days.

To serve, run a sharp knife around the edge of the pan to loosen the cake and remove the side of the pan. Cut the cheesecake into thin wedges with a sharp knife dipped into hot water and wiped dry after each cut.