Let $ R $ be a finitely generated $\mathbb N $-graded algebra domain over a Noetherian
ring and let $ I $ be a homogeneous ideal of $ R $. Given $ P\in Ass (R/I) $ one defines the …

The maximal minors of a matrix of indeterminates are a universal Gröbner basis by a
theorem of Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they are …

Let $ Hilb^ d (A^ 3) $ be the Hilbert scheme of $ d $ points in $ A^ 3$, and let $ T_z $ denote
the tangent space to a point $ z\in Hilb^ d (A^ 3) $. Okounkov and Pandharipande have …

We study several properties of multihomogeneous prime ideals. We show that the
multigraded generic initial ideal of a prime has very special properties, for instance, its …

We consider the problem of computing a grevlex Gröbner basis for the set Fr (M) of minors of
size r of an n× n matrix M of generic linear forms over a field of characteristic zero or large …

We prove that a generic matrix of bounded rank is uniquely recoverable—up to a
permutation of its rows and columns—from an arbitrary permutation of its entries. This can …

The v-function of a graded filtration I={I [k]} k≥ 0 is introduced. Under the assumption that I is
Noetherian, we prove that the v-function v (I [k]) is an eventually quasi-linear function. This …

We study when blowup algebras are $ F $-split or strongly $ F $-regular. Our main focus is
on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a …

In types C and B we construct new families of toric degenerations and Newton--Okounkov
bodies of flag varieties and also of PBW-monomial bases in irreducible representations. The …

We survey results produced from the interaction between methods in prime characteristic
and combinatorial commutative algebra. We showcase results for edge ideals, toric …