Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Divisor class groups and graded canonical modules of multisection rings

Abstract: We describe the divisor class group and the graded canonical module of the multisection ring T(X;D1,…,Ds) for a normal projective variety X and Weil divisors D1,…,Ds on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor.

Linear systems on a special rational surface

Abstract: We study the Hilbert series of a family of ideals J_\phi generated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in P^{n-1}. In the three variable case this is equivalent to studying the dimension of a linear system on a blow up of P^2. The ideals that arise have the points in very special position, but because there are only seven points, we can apply results of Harbourne to obtain the classes of the negative curves. Reducing to an effective, nef divisor and using Riemann-Roch yields a formula for the Hilbert series. This proves the n=3 case of a conjecture of Postnikov and Shapiro, which they later showed true for all n. Postnikov and Shapiro observe that for a family of ideals closely related to J_\phi a similar result often seems to hold, although counterexamples exist for n=4 and n=5. Our methods allow us to prove that for n=3 an analogous formula is indeed true. We close with a counterexample to a conjecture Postnikov and Shapiro make about the minimal free resolution of these ideals.

Mean value estimate for the greatest divisors of number which is prime to a certain integer

Abstract: For a fixed natural square-free number k,the mean value of the greatest divisor of n which is prime to k was studied

Estimates on volumes of homogeneous polynomial spaces

Abstract: In this paper we develop the ``local part'' of our local/global approach to globally valued fields (GVFs). The ``global part'', which relies on these results, is developed in a subsequent paper. We study \emph{virtual divisors} on projective varieties defined over a valued field $K$, as well as \emph{sub-valuations} on polynomial rings over $K$ (analogous to homogeneous polynomial ideals). We prove a Nullstellensatz-style duality between projective varieties equipped with virtual divisors (analogous to projective varieties over a plain field) and certain sub-valuations on polynomial rings over $K$ (analogous to homogeneous polynomial ideals). Our main result compares the \emph{volume} of a virtual divisor on a variety $W$, namely its $(\dim W + 1)$-fold self-intersection, with the asymptotic behaviour of the volume of the dual sub-valuation, restricted to the space of polynomial functions of degree $m$, as $m \rightarrow \infty$.

On a question of Sidorenko

Abstract: For a positive integer $n>1$ denote by $\omega(n)$ the maximal possible number $k$ of different functions $f_1,\dots,f_k:\mathbb{Z}/n\mathbb{Z}\mapsto \mathbb{Z}/n\mathbb{Z}$ such that each function $f_i-f_j,i