Ideals with an assigned initial ideal

The stratum St(J,<) (the homogeneous stratum Sth(J,<) respectively) of a monomial ideal J in a polynomial ring R is the family of all (homogeneous) ideals of R whose initial ideal with respect to the term order < is J. St(J,<) and Sth(J,<) have a natural structure of affine schemes. Moreover they are homogeneous w.r.t. a non-standard grading called level. This property allows us to draw consequences that are interesting from both a theoretical and a computational point of view. For instance a smooth stratum is always isomorphic to an affine space (Corollary 3.6). As applications, in Sec. 5 we prove that strata and homogeneous strata w.r.t. any term ordering < of every saturated Lex-segment ideal J are smooth. For Sth(J,Lex) we also give a formula for the dimension. In the same way in Sec. 6 we consider any ideal R in k[x0,..., xn] generated by a saturated RevLex-segment ideal in k[x,y,z]. We also prove that Sth(R,RevLex) is smooth and give a formula for its dimension.Comment: 14 pages, improved version, some more example

Z-linear Gale duality and poly weighted spaces (PWS)

The present paper is devoted to discussing Gale duality from the Z-linear algebraic point of view. This allows us to isolate the class of Q-factorial complete toric varieties whose class group is torsion free, here called poly weighted spaces (PWS), as an interesting generalization of weighted projective spaces (WPS).Comment: 29 pages: revised version to appear in Linear Algebra and Its Applications. Major changes: the paper has been largely rewritten following refree's comments. In particular, main geometric results have been anticipated giving rise to the motivational Section

A Q\mathbb{Q}--factorial complete toric variety with Picard number 2 is projective

This paper is devoted to settle two still open problems, connected with the existence of ample and nef divisors on a Q-factorial complete toric variety. The first problem is about the existence of ample divisors when the Picard number is 2: we give a positive answer to this question, by studying the secondary fan by means of Z-linear Gale duality. The second problem is about the minimum value of the Picard number allowing the vanishing of the Nef cone: we present a 3-dimensional example showing that this value cannot be greater then 3, which, under the previous result, is also the minimum value guaranteeing the existence of non-projective examples.Comment: 10 pages, 5 figures. Minor changes following the referee's advise: list of notation suppressed, few typos fixed, references updated. Final version to appear in Advances in Geometr

A Q-factorial complete toric variety is a quotient of a poly weighted space

We prove that every Q-factorial complete toric variety is a finite quotient of a poly weighted space (PWS), as defined in our previous work arXiv:1501.05244. This generalizes the Batyrev-Cox and Conrads description of a Q-factorial complete toric variety of Picard number 1, as a finite quotient of a weighted projective space (WPS) \cite[Lemma~2.11]{BC} and \cite[Prop.~4.7]{Conrads}, to every possible Picard number, by replacing the covering WPS with a PWS. As a consequence we describe the bases of the subgroup of Cartier divisors inside the free group of Weil divisors and the bases of the Picard subgroup inside the class group, respectively, generalizing to every Q-factorial complete toric variety the description given in arXiv:1501.05244, Thm. 2.9, for a PWS.Comment: 25+9 pp. Post-final version of our paper published in Ann.Mat.Pur.Appl.(2017),196,325-347: after its publication we realized that Prop.~3.1 contains an error strongly influencing the rest of the paper. Here is a correct revision (first 25 pp.: this version will not be published) and the Erratum appearing soon in Ann. Mat. Pur. Appl. (last 9 pp.) correcting only those parts affected by the erro

On the finiteness and periodicity of the pp--adic Jacobi--Perron algorithm

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to obtain periodic representations for algebraic irrationals, as it is for continued fractions and quadratic irrationals. Since continued fractions have been also studied in the field of $p$--adic numbers $\mathbb Q_p$, also MCFs have been recently introduced in $\mathbb Q_p$ together to a $p$--adic Jacobi--Perron algorithm. In this paper, we address th study of two main features of this algorithm, i.e., finiteness and periodicity. In particular, regarding the finiteness of the $p$--adic Jacobi--Perron algorithm our results are obtained by exploiting properties of some auxiliary integer sequences. Moreover, it is known that a finite $p$--adic MCF represents $\mathbb Q$--linearly dependent numbers. We see that the viceversa is not always true and we prove that in this case infinite partial quotients of the MCF have $p$--adic valuations equal to $-1$. Finally, we show that a periodic MCF of dimension $m$ converges to algebraic irrationals of degree less or equal than $m+1$ and for the case $m=2$ we are able to give some more detailed results

Embedding the Picard group inside the class group: the case of \Q-factorial complete toric varieties

Let $X$ be a \Q-factorial complete toric variety over an algebraic closed field of characteristic $0$. There is a canonical injection of the Picard group ${\rm Pic}(X)$ in the group ${\rm Cl}(X)$ of classes of Weil divisors. These two groups are finitely generated abelian groups; whilst the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of ${\rm Pic}(X)$ in ${\rm Cl}(X)$ is contained in a free part of the latter group.Comment: 18 pages - References update