Curves Dialogue Box With Default Points?

A third option would be to create an action or script.

Although I don’t know if the default can be set, there are now two ways to apply a preset, and the newer way is both faster and more powerful than the older way, so that you could apply your default curve in one click.

The older way is the preset you can save for Curves, just as you can for other adjustment layer types. This is what AxelMatt mentioned. Because it isn’t new it’s been a familiar, reliable way to address this for many years. 

The newer way is the Adjustment Preset. Instead of being tied to a specific adjustment layer type, adjustment presets can include multiple adjustment layers of different types. That’s nice if your typical correction involves a group of different adjustment layers. An adjustment preset also remembers the blending mode of any layer in it, unlike a Curves preset that can only remember the points.

The other advantage of an adjustment preset is it’s faster. What makes the traditional Curves preset slower is you have to add the Curves layer, then choose from its Preset menu. With an adjustment preset, if the Adjustments panel is already open and your preset is visible in the list, you just click it and both the curve and its settings are added in that one step. If you make one of these, drag it to the top of the Adjustments Presets list for instant access, because it’s created at the end of the list where you’ll have to scroll to see it.

You can create a preset. 

Add a curve layer to your image and set the appropriate points on the curve.

Select the curve layer and create a preset.

Create Your Own Photoshop Presets - YouTube