Mathematics

Let V,W be representations of a cyclic group G of prime order p over a field k of characteristic p . The module of covariants k[V,W ] G is the set of G -equivariant polynomial maps V→W , and is a module over k[V ] G . We give a formula for the Noether bound β(k[V,W ] G ,k[V ] G ) , i.e. the minimal degree d such that k[V,W ] G is generated over k[V ] G by elements of degree at most d .

One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear naturally by considering the Rees algebra (blowup) of the base ideal defining the map. Several special cases are obtained as consequences, some of which cover and extend previous results in the literature.

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective spaces. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.

A squarefree monomial ideal is called an f -ideal if its Stanley-Reisner and facet simplicial complexes have the same f -vector. We show that f -ideals generated in a fixed degree have asymptotic density zero when the number of variables goes to infinity. We also provide novel algorithms to construct f -ideals generated in small degrees.

Given an algebraic function field F|K and a place ℘ on K , we prove that the places that are composite with extensions of ℘ to finite extensions of K lie dense in the space of all places of F , in a strong sense. We apply the result to the case of K=R any real closed field and the fixed place on R being its natural (finest) real place. This leads to a new description of the real holomorphy ring of F which can be seen as an analogue to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space M(F) of all $\R$-places of F (places with residue field contained in $\R$), its subspace of all $\R$-places of F that are composite with the natural $\R$-place of R , and the topological space of all R -rational places. Further results about these spaces as well as various classes of relative real holomorphy rings are proven. At the conclusion of the paper the theory of real spectra of rings will be applied to interpret basic concepts from that angle and to show that the space M(F) has only finitely many topological components.

Let F,G in C[x_1,...,x_n] be two polynomials in n variables x_1,...,x_n over the complex numbers field C. In this paper, we prove that if the degree of the Poisson bracket [F,G] is small enough then there are strict constraints for homogeneous components of these polynomials. We also prove that there is a relationship between the homogeneous components of the polynomial F of degrees deg(F)-1 and deg(F)-2 as well some results about divisibility of the homogeneous component of degree deg(F)-1. Moreover we propose, possibly an appropriate, reformulation of the conjecture of Yu regarding the estimation of the Poisson bracket degree of two polynomials.

Let G be a group acting via ring automorphisms on an integral domain R. A ring-theoretic property of R is said to be G -invariant, if R G also has the property, where R G ={r∈R | σ(r)=r for all σ∈G}, the fixed ring of the action. In this paper we prove the following classes of rings are invariant under the operation R→ R G : locally pqr domains, Strong G-domains, G-domains, Hilbert rings, S -strong rings and root-closed domains. Further let P be a ring theoretic property and R⊆S be a ring extension. A pair of rings (R,S) is said to be a P -pair, if T satisfies P for each intermediate ring R⊆T⊆S. We also prove that the property P descends from (R,S)→( R G , S G ) in several cases. For instance, if P= Going-down, Pseudo-valuation domain and "finite length of intermediate chains of domains", we show each of these properties successfully transfer from (R,S)→( R G , S G ).

A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic n×m matrix indexed by the facets of a simplicial complex. We consider the more general notion of an r -DFI, which is generated by a subset of r -minors of a generic matrix indexed by the facets of a simplicial complex for some 1≤r≤n . We define and study so-called lcm-closed and interval DFIs, and show that the minors parametrized by the facets of Δ form a reduced Gröbner basis with respect to any diagonal term order in both of these cases. We also see that being lcm-closed generalizes conditions previously introduced in the literature, and conjecture that in the case r=n , lcm-closedness is necessary for being a Gröbner basis. We also give conditions on the maximal cliques of Δ ensuring that lcm-closed and interval DFIs are Cohen-Macaulay. Finally, we conclude with a variant of a conjecture of Ene, Herzog, and Hibi on the Betti numbers of certain types of r -DFIs, and provide a proof of this conjecture for Cohen-Macaulay interval DFIs.

Knutson and Miller (2005) established a connection between the anti-diagonal Gröbner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a geometrically-natural explanation for the appearance of the combinatorially defined Schubert polynomials as representatives of Schubert classes. Recently, Hamaker, Pechenik, and Weigandt proposed a similar connection between diagonal degenerations of matrix Schubert varieties and bumpless pipe dreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono. Hamaker, Pechenik, and Weigandt described new generating sets of the defining ideals of matrix Schubert varieties and conjectured a characterization of permutations for which these generating sets are diagonal Gröbner bases. They proved special cases of this conjecture and described diagonal degenerations of matrix Schubert varieties in terms of bumpless pipe dreams in these cases. The purpose of this paper is to prove the general conjecture. The proof uses a connection between liaison and geometric vertex decomposition established in earlier work with Rajchgot.

Differential modules over a commutative differential ring R which are finitely generated projective as ring modules, with differential homomorphisms, form an additive category, so their isomorphism classes form a monoid. We study the quotient monoid of this monoid by the submonoid of isomorphism classes of free modules with component wise derivation. This quotient monoid has the reduced K group of R (ignoring the derivation) as an image and contains the reduced K group of the constants of R as its subgroup of units. This monoid provides a description of the isomorphism classes of differential projective R modules up to an equivalence.