Gerundial (Noun’s or Possessive Pronoun + Gerund);
Do you mind John’s smoking in the room?
I insist on Mary’s going there.
The ‘possessive’ form is considered to be more literary and elegant;
Half-Gerundial (Noun in the Common case or Personal Pronoun in the Objective Case + Gerund).
I insist on Mary going there.
the ‘objective’ form is found mainly in the spoken language, “where it is probably just as common as the possessive form”
(W. Stannard Allen)
Constructions with the Participle. 1) The objective – participle construction; 2) The subjective – participle construction; 3) The subjective – absolute participle construction; 4) The absolute participle construction