Curing spacetime singularities by a higher-curvature gravitational Lagrangian It is well known that the self-energy of a point charge is diverging in the classical (Maxwell) electrodynamics. In 1930's, Born and Infeld introduced their nonlinear electrodynamics (with a Lagrangian involving higher-order corrections to the standard Maxwell Lagrangian when expanded in the field strength) in order to make the self-energy of point charges finite. 


*

*Would it be possible that (at least some) spacetime singularities
could be cured by a similar mechanicsm, i.e. by introducing suitable
higher-curvature terms in the gravitational Lagrangian?


I know that such higher-curvature corrections to the Einstein-Hilbert Lagrangian appear in the string theory effective action, therefore: 


*

*Is this the way how people expect the string theory to get rid of the
spacetime singularities or is there some different mechanism behind
it?

 A: *

*The low energy limit of string theory is supergravity which does have higher derivative corrections to the Einstein Hibert action, but in general it neither gets "rid" of singularities nor there is a necessity to get "rid" of them. Note that supergravity is the low energy effective field theory limit, if you wish to consider effects at Planck scale such as singularities you will need to take into account stringy corrections. These stringy corrections are best understood in the AdS/CFT setup, where you have a dual field theoretic description which doesn't have singularities. 

*I will talk specifically about black holes here. If you take a bunch of strings at very strong coupling and excite them to a very high energy above the Hagedorn temperature, you get the Bekenstein Hawking entropy counting and consequently you are looking at black holes. In the AdS/CFT setup you get black holes by exciting the dual gauge theoretic degrees of freedom above the Hagedorn temperature. There is no problem here which the existence of singularities pose.

*As @Void mentioned in the comments, Jimenez et al discuss some possible non singular black hole solutions in detail. Solutions such as these might or might not be realized in a string-theoretic setup.

*There is another question which is what curing singularities buys you in quantum gravity. There are important effects like the information paradox and bags of gold paradox which don't depend on black hole singularities, but rather arise due to the existence of the horizon itself. All black hole paradoxes in quantum gravity are concerned with the interior degrees of freedom in the black hole, and can be formulated on "nice spacelike slices" which stay away from singularities everywhere. Removing singularities classically in your proposed fashion doesn't cure the problem at all here, but gives you a different problem altogether, which is basically that now you need to find an UV completion for the low energy singularity free theory, and there is no guarantee that a stringy setup will be the UV completion for your class of higher derivative singularity free theories. 
