Valuative stability of polarised varieties

The GIT-stability of Polarised Varieties via discrepancy

We study the GIT stability of polarised varieities, in the language of the discrepancy of singularities, developed along the Mori Program. We give a new geometric numerical invariant of polarised varities, which is directly related to the discrepancy of singularities, whose negativity destabilises the polarised varieties. We relate this new invariant to the invariant which was used in J.Shah’s ...

Moduli spaces of polarised symplectic O’Grady varieties and Borcherds products

We study moduli spaces of O’Grady’s ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both t...

Valuative Arf Characteristic of Singularities

The proof by Hironaka [5] of resolution of singularities of algebraic varieties over fields of characteristic zero raised the problem of classifying singularities by looking at the resolution process. Thus, equisingularity of plane curve singularities was introduced and developed by Zariski in [15] by showing that the combinatorics of the resolution processes is equivalent data to Puiseux invar...

The Valuative Theory of Foliations

Valuative invariants for polymatroids

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-JiaReiner quasi-symmetric function, and the invariant G introduced by the first author, are valuative. In this paper we construct the Z-modules of all Z-valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their unlabeled counterparts, the Z-modules of valuat...