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We consider the moduli space of log smooth pairs formed by a cubic surface and an anticanonical divisor. We describe all compactifications of this moduli space which are constructed using Geometric Invariant Theory and the anticanonical polarization. The construction depends on a weight on the divisor. For smaller weights the stable pairs consist of mildly singular surfaces and very singular divisors. Conversely, a larger weight allows more singular surfaces, but it restricts the singularities on the divisor. The one-dimensional space of stability conditions decomposes in a wall-chamber structure. We describe all the walls and relate their value to the worst singularities appearing in the compactification locus. Furthermore, we give a complete characterization of stable and polystable pairs in terms of their singularities for each of the compactifications considered.

We study the GIT compactifications of pairs formed by a hypersurface and a hyperplane. We provide a general setting to characterize all polarizations which give rise to different GIT quotients. Furthermore, we describe a finite set of one-parameter subgroups sufficient to determine the stability of any GIT quotient. We characterize all maximal orbits of non stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs. Our construction gives natural compactifications of the space of log smooth pairs for Fano and Calabi-Yau hypersurfaces.GIT n,d,t corresponding to values t = t i and to any t ∈ (t i , t i+1 ).Theorem 1.1. Let S n,d be the set of one-parameter subgroups in Definition 3.1. All GIT walls {t 0 , . . . , t n,d } correspond to a subset of the finite set − m, λ x i , λ m is a monomial of degree d, 0 i n + 1, λ ∈ S n,d (1)and they are contained in the interval [0, t n,d ] where t n,d = d n+1 . Every pair (X, H) has an interval of stability [a, b] with a, b ∈ {t 0 , . . . , t n,d }. Namely, (X, H) is t-semistable if and only if t ∈ [t i , t j ] for some walls t i , t j . If (X, H) is t-stable for some t then (X, H) is t-stable if and only if t ∈ (t i , t j ).Corollary 1.2. Assume that the ground field is algebraically closed with characteristic 0 and that the locus of stable points is not empty and d 3. Then dim M GIT n,d,t =

We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in three dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second component of this Hilbert scheme appears. Along the way, we prove that the principal component of this Hilbert scheme is a double blow up with smooth centers of a Grassmannian, exhibiting a completely different proof of this known result by Avritzer and Vainsencher. This description allows us to compute the cone of effective divisors of this component.

The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of Kähler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting' to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from Geometric Invariant Theory. More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients. GIT d of degree d hypersurfaces in P n. The first statement of the above result is a special case of the following natural expectation, at least when the automorphism group has no non-trivial characters. Conjecture 1.3. Let X be a K-polystable Fano variety. Then for any sufficiently large and divisible l ∈ N, there exists β 0 = β 0 (l, X) such that for all D ∈ | − lK X | and β ∈ (β 0 , 1), the pair (X, (1 − β)D) is log Kpolystable if and only if [D] ∈ P(H 0 (−lK X)) is GIT-polystable for the natural representation of Aut(X) on H 0 (−lK X). Moreover, the Gromov-Hausdorff compactification M K X,l,β of the above pairs (X, (1 − β)D) is homeomorphic to the GIT quotient P(H 0 (−lK X)) ss //Aut(X). Note that the automorphism group Aut(X) of a K-polystable Fano is always reductive. If X is smooth or a Gromov-Hausdorff degeneration, this follows from by Matsushima's obstruction [Mat57] and Chen-Donaldson-Sun [CDS15]. Otherwise, this has been very recently established in [ABHLX20] via purely algebro-geometric techniques Once its different steps are established, the application of the moduli continuity method (the proof of theorems 1.1 and 1.2) is just a couple of paragraphs. However, in order to keep the reader focused on the main goal of the paper, we recall here what the different steps of the moduli continuity method are, noting that each of them requires involved technical proofs to accomplish them. Suppose that we have a Gromov-Hausdorff