Research

Residual Intersections: 

Historically, linkage (liaison) began in the nineteenth century as a tool to classify curves in projective spaces. The idea was to start with a curve, say in P^3, and look at its residual in a complete intersection. Since complete intersections are in some sense the simplest curves, it turns out that a lot of information can be carried over from a curve to its residual.

Liaison has been used very extensively in the literature as a means of studying curves (or higher dimensional varieties). Furthermore, beginning with the paper of Peskine and Szpiro 1970's, liaison has attracted a great deal of attention as a subject itself, rather than simply a tool to study projective varieties. Nowadays, there exist a considerable number of fields branching from linkage theory, including Deficiency Modules, Linkage of Modules, and Residual Intersections.

Below, you can find some of my works on this topic:

1. S.H. Hassanzadeh, Residual Intersections admit free approach, (2024).

2. S.H .Hassanzadeh and K. Vasconcellos, Deformation of residual intersections, (2024), Arxiv 

3. Bouça, Vinicius; Hassanzadeh, S. Hamid Residual intersections are Koszul-Fitting ideals. Compos. Math. 155 (2019), no. 11, 2150–2179. The arxiv version is here.

4. Hassanzadeh, Seyed Hamid; Naéliton, Jose Residual intersections and the annihilator of Koszul homologies. Algebra Number Theory 10 (2016), no. 4, 737–770. The arxiv version is here.

5. Hassanzadeh, Seyed Hamid Cohen-Macaulay residual intersections and their Castelnuovo-Mumford regularity. Trans. Amer. Math. Soc. 364 (2012), no. 12, 6371–6394. The arxiv version is here. 

6. Hassanzadeh, S. H.; Shirmohammadi, N.; Zakeri, H. Linkage and strongly generalized Cohen-Macaulay ideals. Algebra Colloq. 17 (2010), no. 1, 153–160.

7. Dibaei, Mohammad T.; Gheibi, Mohsen; Hassanzadeh, S. H.; Sadeghi, Arash Linkage of modules over Cohen-Macaulay rings. J. Algebra 335 (2011), 177–187. The arxiv version is here.

Rational Maps: 

Rational maps are among the central objects in Algebraic Geometry. From a commutative algebra point of view, a rational map F: X ⊆ P^n --->P^m is defined by m+1 forms f = f0,···fm, of the elements in the coordinate ring of X, of the same degree in n+1 variables not all vanishing. One of the most subtle problems in algebraic geometry is to decide when a rational map is birational which leads to birational classification of varieties. Cremona maps are special cases of birational maps which are among the most interesting objects in many lines of research.

Below, you can find some of my works on this topic:

1. S.H.Hassanzadeh, M.Mostafazadehfard, Bir(X)_d is constructible, Proc. Amer. Math. Soc. 152 (2024), no.5, 1841-1856,  arXiv:2208.12333 

2. M. Chardin. S.H. Hassanzadeh, A. Simis,  Degree of Rational Maps versus Syzygies, J. Algebra, Volume 573, 1 May 2021, Pages 641-662,  arXiv:2007.13017 [math.AC].

3. Hassanzadeh, S. H.; Simis, A. Bounds on degrees of birational maps with arithmetically Cohen-Macaulay graphs. J. Algebra 478 (2017), 220–236. The arxiv version is here. 

4. Botbol, Nicolás; Busé, Laurent; Chardin, Marc; Hassanzadeh, Seyed Hamid; Simis, Aron; Tran, Quang Hoa Effective criteria for bigraded birational maps. J. Symbolic Comput. 81 (2017), 69–87. The arxiv version is here. 

5.  Hassanzadeh, Seyed Hamid; Simis, Aron Implicitization of de Jonquières parametrizations. J. Commut. Algebra 6 (2014), no. 2, 149–172.  The arxiv version is here. 

6. Hassanzadeh, Seyed Hamid; Simis, Aron Plane Cremona maps: saturation and regularity of the base ideal. J. Algebra 371 (2012), 620–652.  The arxiv version is here.

7. Doria, A. V.; Hassanzadeh, S. H.; Simis, A. A characteristic-free criterion of birationality. Adv. Math. 230 (2012), no. 1, 390–413.  The arxiv version is here.

Other Topics:

Commutative Algebra is a very active and vast field of research.  It has strong connections with other fields such as Algebraic Geometry, Combinatorics, Topology,  and Statistics. 

Below, you can find some of my works on different topics

1. M.Chardin, S.H.Hassamzadeh,C.Polini, A.Simis and B.Ulrich, BOUNDS ON DEGREES OF VECTOR FIELDS, (2024), Arxiv  

2. W.A. da Silva, S. H. Hassanzadeh, A. Simis, Bounds for the degree and Betti sequences along a graded resolution, (2024), Journal of Algebra and its Applications

3. V. Bouça, T. Fiel, S.H.Hassanzadeh, J. Naeliton, Buchsbaum-Rim multiplicity: A Koszul homology description, (2023) Volume 633, 1 November 2023, Pages 754-772, Journal Of Algebra

4.  Boocher, Adam; Hassanzadeh, S. Hamid; Iyengar, Srikanth B. Koszul algebras defined by three relations. Homological and computational methods in commutative algebra, 53–68, Springer INdAM Ser., 20, Springer, Cham, 2017. The arxiv version is here. 

5.  Borna, Keivan; Hassanzadeh, S. H. Ideals whose first two Betti numbers are close. Algebra Colloq. 22 (2015), no. 4, 671–676. The arxiv version is here. 

6. Hassanzadeh, S. H.; Jahangiri, M.; Zakeri, H. Asymptotic behavior and Artinian property of graded local cohomology modules. Comm. Algebra 37 (2009), no. 11, 4095–4102. The arxiv version is here. 

7. Hassanzadeh, S. H.; Vahidi, Alireza On vanishing and cofiniteness of generalized local cohomology modules. Comm. Algebra 37 (2009), no. 7, 2290–2299. The arxiv version is here.

8.  Hassanzadeh, S. H.; Shirmohammadi, N.; Zakeri, H. A note on quasi-Gorenstein rings. Arch. Math. (Basel) 91 (2008), no. 4, 318–322. The arxiv version is here. 

Macaulay-2:

The computational approach in commutative algebra and algebraic geometry is an indispensable part of the research in the new era.

Here are some methods that I wrote during my research:

1. The package on the rational maps, joint with Karl Schwede. 

2. The method clearDegree: given a rational map, the method determines the degree of a representative of grade at least 2.

3. The method findDominantMap: given varieties X=Proj(R) and Y=Proj(S) and a degree d: it looks among all of the (monomial, or binomial) rational maps from X to Y of clear degree  d returns some of the birational ones. 

4. The method findBirationalMap: given varieties X=Proj(R) and Y=Proj(S) and a degree d: it looks among all of the (monomial) rational maps from X to Y and returns some of the birational ones. When you download this method you must download clearDegree.m2 and permutationWellDefined.m2

5. The method kitt: given two ideals $a\subset I$ in a commutative ring R. This method computes the ideal Kitt(a,I) defined in the paper [Bouça, Vinicius; Hassanzadeh, S. Hamid Residual intersections are Koszul-Fitting ideals. Compos. Math. 155 (2019), no. 11, 2150–2179. The arxiv version is here. ] concerning the ideal J=a:I, some properties of Kitt(a,I) are the following: 

         1. Kitt(a,I)\subset J,   2.rad(Kitt(a,I))=rad(J), 3. If R is CM, J is an s-residual intersection and I satisfies SD1 then J=Kitt(a,I,) 4. 4. If R is CM, J is an s-residual intersection and s<ht(I)+2 then Kitt(a,I)=J 5. If R is CM , J is an s-residual intersection , I is weakly (s-2)-residually S2 and G_s then Kitt(a,I)=J [Tarasova]

6. The method eliminationIndex: to some extent computes the distance between a variety and P^n .