We study the relationship between geometric properties of toric varieties and combinatorial properties of the corresponding lattice polytopes. In particular, we give a bound for a very ample lattice polytope to be k-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties. We also give a new combinatorial proof for a special case of Reider's Theorem for smooth toric surfaces.

<p>Let S be a polynomial ring K[x1,...,xn] over a field K and let F be a non-negatively graded free module over S generated by m basis elements. In this thesis, we study four kinds of upper bounds: degree bounds for Gröbner bases of submodules of F, bounds for arithmetic degrees of S-ideals, regularity bounds for radicals of S-ideals, and Stillman bounds. </p> <p><br></p> <p>Let M be a submodule of F generated by elements with degrees bounded above by D and dim(F/M)=r. We prove that if M is graded, the degree of the reduced Gröbner basis of M for any term order is bounded above by 2[1/2((Dm)^{n-r}m+D)]^{2^{r-1}}. If M is not graded, the bound is 2[1/2((Dm)^{(n-r)^2}m+D)]^{2^{r}}. This is a generalization of bounds for ideals in a polynomial ring due to Dubé (1990) and Mayr-Ritscher (2013).</p> <p><br></p> <p>Our next results are concerned with a homogeneous ideal I in S generated by forms of degree at most d with dim(S/I)=r. In Chapter 4, we show how to derive from a result of Hoa (2008) an upper bound for the regularity of sqrt{I}, which denotes the radical of I. More specifically we show that reg(sqrt{I})<= d^{(n-1)2^{r-1}}. In Chapter 5, we show that the i-th arithmetic degree of I is bounded above by 2*d^{2^{n-i-1}}. This is done by proving upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type.</p> <p><br></p> <p>In the last chapter, we explain our progress in attempting to make Stillman bounds explicit. Ananyan and Hochster (2020) were the first to show the existence of Stillman bounds. Together with G. Caviglia, we observe that a possible way of making their results explicit is to find an effective bound for an invariant called D(k,d) and supplement it into their proof. Although we are able to obtain this bound D(k,d) and realize Stillman bounds via an algorithm, it turns out that the computational complexity of Ananyan and Hochster's inductive proof would make the bounds too large to be meaningful. We explain the bad behavior of these Stillman bounds by giving estimates up to degree 3.</p>

<p>Consider the standard graded polynomial ring in $n$ variables over a field $k$ and fix the Hilbert function of a homogeneous ideal. In the nineties Bigatti, Hulett, and Pardue showed that the Hilbert scheme consisting of all the homogeneous ideals with such a Hilbert function contains an extremal point which simultaneously maximizes all the graded Betti numbers. Such a point is the unique lexsegment ideal associated to the fixed Hilbert function.</p> <p> For such a scheme, we consider the individual strata defined by all ideals with Castelnuovo-Mumford regularity bounded above by <em>m</em>. In 1997 Mall showed that when <em>k </em>is of characteristic 0 there exists an ideal in each nonempty strata with maximal possible Betti numbers among the ideals of the strata. In chapter 4 of this thesis we provide a new construction of Mall's ideal, extend the result to fields of any characteristic, and show that these ideals have other extremal properties. For example, Mall's ideals satisfy an equation similar to Green's hyperplane section theorem.</p> <p> The key technical component needed to extend the results of Mall is discussed in Chapter 3. This component is the construction of a new invariant called the distraction-generic initial ideal. Given a homogeneous ideal <em>I C S</em> we construct the associated distraction-generic initial ideal, D-gin_< (<em>I</em>), by iteratively computing initial ideals and general distractions. The result is a monomial ideal that is strongly stable in any characteristic and which has many properties analogous to the generic initial ideal of <em>I</em>.</p>

Seja (R;m) um anel Noetheriano local e R \'CONTÉM\' \'iota IND. 1\' \'CONTÉM\' \'iota IND. 2\' \'CONTÉM ... uma filtração de ideais de R. Podemos então construir a álgebra graduada F(\'\\Im) := \'SOMA DIRETA IND. n > OU = 0 POT. \'iota IND. n / \'m \'iota IND. n\', chamada de fibra especial. Esta tese objetiva a pesquisa deste anel. Investigamos sobre a sua propriedade de ser Gorenstein e a sua regularidade de Castelnuovo-Mumford. Outro objetivo, é generalizarmos o teorema de Risler-Teissier (sobre multiplicidades mistas) para o caso de filtrações de Hilbert / Let (R;m) be a Noetherian local ring and R \'CONTAINS\' \'iota IND. 1\' \'CONTAINS\' \'iota IND. 2\' \'CONTAINS\' ... a filtration of ideals in R. We may then construct the graded algebra F(\\Im) := \'DIRECT SUM\' IND. n > OR = \'0 POT. \'iota\' IND. n / \'m \'iota IND. n\' , which is called fiber cone. This thesis has the goal to research about this graded ring. We investigate its Gorenstein property and its Castelnuovo-Mumford regularity. Another aim is to generalize the Risler-Teissiers theorem (about mixed multiplicities) for the case of Hilbert filtration

Le sujet de cette thèse est l'étude d'idéaux monomiaux de l'anneau de polynômes S qui ont une résolution linéaire. D'après un résultat remarquable de Bayer et Stilman et en utilisant la polarisation, la classification des idéaux monomiaux ayant une résolution linéaire est équivalente à la classification des idéaux monomiaux libres de carrés ayant une résolution linéaire. Pour cette raison dans cette thèse nous considérons seulement le cas d'idéaux monomiaux libres de carrés. De plus, le théorème de Eagon-Reiner établit une dualité entre les idéaux monomiaux libres de carrés ayant une résolution linéaire et les idéaux monomiaux libres de carrés Cohen-Macaulay, ce qui montre que le problème de classification des idéaux monomiaux libres de carrés ayant une résolution linéaire est très difficile. Nous rappelons que les idéaux monomiaux libres de carrés sont en correspondance biunivoque avec les complexes simpliciaux d'une part, et d'autre part avec les clutters. Ces correspondances nous motivent pour utiliser les propriétés combinatoires des complexes simpliciaux et des clutters pour obtenir des résultats algébriques. La classification des idéaux monomiaux libres de carrés ayant une résolution linéaire engendrés en degré 2 a été faite par Froberg en 1990. Froberg a observé que l'idéal des circuits d'un graphe G a une résolution 2-linéaire si et seulement si G est un graphe de cordes, i.e. il n'a pas de cycles minimaux de longueur plus grande que 4. Dans [Em, ThVt, VtV, W] les auteurs ont partiellement généralisé les résultats de Froberg à des idéaux engendrés en degré >2. Ils ont introduit plusieurs définitions de clutters de cordes et démontré que les idéaux de circuits correspondant ont une résolution linéaire. Nous pouvons voir les cycles du point de vue topologique, comme la triangulation d'une courbe fermée, dans cette thèse nous utiliserons cette idée pour étudier des clutters associés à des triangulation de pseudo-manifolds en vue d'obtenir une généralisation partielle des résultats de Froberg à des idéaux engendrés en degré >2. Nous comparons notre travail à ceux de [Em, ThVt, VtV, W]. Nous présentons nos résultats dans le chapitres 4 et 5. / In this thesis, we study square-free monomial ideals of the polynomial ring S which have a linear resolution. By remarkable result of Bayer and Stilman [BS] and the technique of polarization, classification of ideals with linear resolution is equivalent to classification of square-free monomial ideals with linear resolution. For this reason, we consider only square-free monomial ideals in S. However, classification of square-free monomial ideals with linear resolution seems to be so difficult because by Eagon-Reiner Theorem [ER], this is equivalent to classification of Cohen-Macaulay ideals. It is worth to note that, square-free monomial ideals in S are in one-to-one correspondence to Stanley-Reisener ideals of simplicial complexes on one hand and the circuit ideal of clutters from another hand. This correspondence motivated mathematicians to use the combinatorial and geometrical properties of these objects in order to get the desired algebraic results. Classification of square-free monomial ideals with 2-linear resolution, was successfully done by Froberg [Fr] in 1990. Froberg observed that the circuit ideal of a graph G has a 2-linear resolution if and only if G is chordal, that is, G does not have an induced cycle of length > 3. In [Em, ThVt, VtV, W] the authors have partially generalized the Fr¨oberg's theorem for degree greater than 2. They have introduced several definitions of chordal clutters and proved that, their corresponding circuit ideals have linear resolutions. Viewing cycles as geometrical objects (triangulation of closed curves), in this thesis we try to generalize the concept of cycles to triangulation of pseudo-manifolds and get a partial generalization of Froberg's theorem for higher dimensional hypergraphs. All the results in Chapters 4 and 5 and some results in Chapter 3 are devoted to be the original results.

Seja (R;m) um anel Noetheriano local e R \'CONTÉM\' \'iota IND. 1\' \'CONTÉM\' \'iota IND. 2\' \'CONTÉM ... uma filtração de ideais de R. Podemos então construir a álgebra graduada F(\'\\Im) := \'SOMA DIRETA IND. n > OU = 0 POT. \'iota IND. n / \'m \'iota IND. n\', chamada de fibra especial. Esta tese objetiva a pesquisa deste anel. Investigamos sobre a sua propriedade de ser Gorenstein e a sua regularidade de Castelnuovo-Mumford. Outro objetivo, é generalizarmos o teorema de Risler-Teissier (sobre multiplicidades mistas) para o caso de filtrações de Hilbert / Let (R;m) be a Noetherian local ring and R \'CONTAINS\' \'iota IND. 1\' \'CONTAINS\' \'iota IND. 2\' \'CONTAINS\' ... a filtration of ideals in R. We may then construct the graded algebra F(\\Im) := \'DIRECT SUM\' IND. n > OR = \'0 POT. \'iota\' IND. n / \'m \'iota IND. n\' , which is called fiber cone. This thesis has the goal to research about this graded ring. We investigate its Gorenstein property and its Castelnuovo-Mumford regularity. Another aim is to generalize the Risler-Teissiers theorem (about mixed multiplicities) for the case of Hilbert filtration

Since the early 1990s, a classical object in commutative algebra has been the study of binomial ideals. A widely-investigated class of binomial ideals is the one containing those generated by a subset of 2-minors of an (m x n)-matrix of indeterminates. This thesis is devoted to illustrate some algebraic and homological properties of two classes of ideals of 2-minors: binomial edge ideals and polyomino ideals. Binomial edge ideals arise from finite graphs and their appeal results from the fact that their homological properties reflect nicely the combinatorics of the underlying graph. First, we focus on the binomial edge ideals of block graphs. We give a lower bound for their Castelnuovo-Mumford regularity by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a linear time algorithm to compute Castelnuovo-Mumford regularity and Krull dimension of binomial edge ideals of block graphs. Secondly, we consider some classes of Cohen-Macaulay binomial edge ideals. We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs. Polyomino ideals arise from polyominoes, plane figures formed by joining one or more equal squares edge to edge. It is known that the polyomino ideal of simple polyominoes is prime. We consider multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14 the above condition is also sufficient. Lastly, we present an infinite class of prime polyomino ideals.

My PhD-project has two main research directions. The first direction is on partial regularities which we define as refinements of the Castelnuovo-Mumford regularity. Main results are: relationship of partial regularities and related invariants, like the a-invariants or the Castelnuovo-Mumford regularity of the syzygy modules; algebraic properties of partial regularities via a filter-regular sequence or a short exact sequence; generalizing a well-known result for the Castelnuovo-Mumford regularity to the case of partial regularities of stable and squarefree stable monomial ideals; finally extending an upper bound proven by Caviglia-Sbarra to partial regularities. The second direction of my project is to develop a theory on monomial preorders. Many interesting statements from the classical theory of monomial orders generalize to monomial preorders. Main results are: a characterization of monomial preorders by real matrices, which extends a result of Robbiano on monomial orders; secondly, leading term ideals with respect to monomial preorders can be studied via flat deformations of the given ideal; finally, comparing invariants of the given ideal and the leading term ideal with respect to a monomial preorder.