Maksym Fedorchuk

Rigidity and polynomial invariants of convex polytopes

We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obtain sharp upper and lower bounds on the degree of these polynomial relations. In a special case of regular bipyramid we obtain explicit formulae for some of these relations. We conclude with a proof of Robbins Conjecture [R2] on the degree of generalized Heron polynomials.

Finite Hilbert stability of (bi)canonical curves

We prove that a generic canonically or bicanonically embedded smooth curve has semistable mth Hilbert points for all m≥2. We also prove that a generic bicanonically embedded smooth curve has stable mth Hilbert points for all m≥3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with

Ample divisors on moduli spaces of pointed rational curves

\mathbb{G}_{m}

GIT semistability of Hilbert points of Milnor algebras

-action, namely the canonically embedded balanced ribbon and the canonically embedded balanced doubleA2k+1-curve. In the bicanonical case, we prove finite Hilbert stability of special hyperelliptic curves, namely Wiman curves. Finally, we give examples of canonically embedded smooth curves whose mth Hilbert points are non-semistable for low values of m, but become semistable past a definite threshold.

Toward GIT stability of syzygies of canonical curves

The second author was partially supported by a Clay Mathematics Institute Liftoff Fellowship during the preparation of this paper.

Second flip in the Hassett–Keel program: existence of good moduli spaces

We study GIT semistability of Hilbert points of Milnor algebras of homogeneous forms. Our first result is that a homogeneous form F in n variables is GIT semistable with respect to the natural

Alternate Compactifications of Moduli Spaces of Curves

{{\mathrm{SL}}}(n)

\bar{M}_g

SL(n)-action if and only if the gradient point of F, which is the first non-trivial Hilbert point of the Milnor algebra of F, is semistable. We also prove that the induced morphism on the GIT quotients is finite, and injective on the locus of stable forms. Our second result is that the associated form of F, also known as the Macaulay inverse system of the Milnor algebra of F, and which is apolar to the last non-trivial Hilbert point of the Milnor algebra, is GIT semistable whenever F is a smooth form. These two results answer questions of Alper and Isaev.