 numbers of projective closer of the affine monomial curves. And this is joint work with professor in Nazi group and Dr. Jaydeep Shah. So, let us start. So, first I will discuss about the what is our main objective of this talk. So, numerical semi-groups defines the numerical semi-group rings. And these are the important object in the study of the singularities of algebraic curves. And these semi-groups defines the that class of curve is called the affine monomial curves. And the higher analog of numerical semi-group is called the affine semi-groups. And these semi-groups are also important for the study of the singularity in commutative algebra and geometry. So, main objective of this work is to understand the particular class of affine semi-group in N2 which defines the projective closer of affine monomial curves. And the study of projective closer of affine monomial curves have been done by Burmezo and Herzog and many more. And they mainly focused on the Cohen-McAule properties and the Betty numbers of the projective closer. And in this work we study those property of the projective closer which preserve after the projective closer of affine monomial curve and their associated semi-groups. And we want to explore those property of affine monomial curves that preserve after taking the projective closer. So, let discuss some background for the talk. So, numerical semi-group is a non-empty subset of a non-negative teacher's natural such that it is a sub monoid of natural number and its complement is finite. And this is the generalization of these semi-groups discussed in the Ompreka's talk. And the x is said to be a generating set of numerical semi-group gamma if the element can be written as the linear combination of the element of x over natural number. And the minimal generating set x is said to be if no proper subset of x generates the gamma. And the cardinality of that x is called the embedding dimension of the numerical semi-group. So, what is affine monomial curve? So, affine monomial curve is if you take a curve parameterized by the t to the power n 1 to t to the power n r where n 1 to n r generates a numerical semi-group then that image gamma is called the affine monomial curves. And the defining ideal of that curve is given by the homomorphism eta defined by the homomorphism eta and the coordinate ring is a k quotient k x 1 to x r quotiented by p gamma is called the homogeneous coordinate ring of the affine monomial curve c gamma. And the cruel dimension of this affine monomial curve is 1 and then what is projected closer of affine monomial curve. So, if you choose a particular semi-group which is generated by n i to n r comma n r minus n i and if you parameterize the curve using this semi-group then that curve is called the projective closer of affine monomial curve. And the defining ideal of the projective closer is given by the homomorphism eta h and if you see the defining ideal of p gamma closer is the nothing but the homogenization of the defining ideal of affine monomial curve. And the cruel dimension of this projective closer the homogeneous coordinate ring of this curve is 2. So, this need not be a Cohen Macaulay, but affine monomial curves are always Cohen Macaulay. So, for understanding the how the projective closer behaves we took the three important family which have in literature many more behavior and all these family have one strong resemblance that all are in affine dimension four space and there is no upper bond on the minimal generating set of the defining ideal of these curves. And we investigated the Cohen Macaulay property of the projective closer of these curves. So, if you see the this table in the first row you have seen the you can see the Betty number of the affine affine and the Betty number of the projective closer is same and the Cohen Macaulay is also but for Bezinski the Betty number for affine curve and Betty number for projective closer is different even the last Betty number for Bezinski curve is unbounded, but for projective closer it is one and similarly for Arsalan curve the Betty numbers are totally different. So, gluing what is gluing of numerical semi group. So, this technique is used for creating example of local one-dimensional Gorenstein curve and complete intersection curve. So, gluing is defined as if gamma 1 and gamma 2 are two numerical semi group minimally generated by these number. And if you choose p and q are two non-generating element from gamma 1 and gamma 2 such that their GCD is one and if you take the new numerical semi group which is generated by q m 1 to q m l and p n 1 to p n l then this semi group is called the gluing of the semi group of gamma 1 and gamma 2. So, we want to so projective closer need not be Cohen Macaulay. So, we want to create an example using this technique when Cohen Macaulay projective closer. So, if gamma 1 and gamma 2 are of numerical semi groups such that their projective closer is Cohen Macaulay then the what happened for if gamma is a gluing of gamma 1 and gamma 2 then is a projective closer of c gamma projective closer of c gamma is Cohen Macaulay or not. So, in general it is not Cohen Macaulay. So, under what condition we give on the gluing such that their projective closer is become Cohen Macaulay. So, we define one special gluing which is called star gluing. So, if you take if you restrict the condition on the non generating element p and q and where p is the multiple of the last generating element of gamma 1 and q is any non generating element of gamma 2 with some condition then we prove that if gamma is a star gluing of gamma 1 and gamma 2 such that their projective closer is Cohen Macaulay or Gordon Stein then the projective closer of the star gluing of star gluing of gamma 1 and gamma 2 is also Cohen Macaulay and Gordon Stein. So, from this technique we can create an infinite example of Cohen Macaulay projective closer. So, next property which we want to study is the Betty how the Betty numbers we have for projective closer and for projective closer. So, using Grobner's basis technique we can we prove that if gamma is a numerical semi group such that its projective closer is arithmetically Cohen Macaulay and if there exists a minimal Grobner basis of defining ideal with respect to some some monomial ordering such that last variable axon belongs to support of all non-homogeneous element then the Betty number of affine monomial curve and the Betty number of its projective closer is same and from that Betty numbers of affine monomial curve associated to arithmetic sequence is given by Gaminis and Saint Gupta and Srinivasan and from our theorem we we proved that the Betty number for the projective closer associated to arithmetic sequence is also same as the Betty numbers of the affine monomial curves. So, next we want to study the how the Betty numbers Betty number projective closer behaves in the case of the simple gluing of numerical semi group and what are we define this simple gluing if gamma is a numerical semi group minimally generated by these numbers and D is any non-generating element then the new numerical semi group which is generated by these numbers is called the simple gluing of gamma gamma and this this simple gluing is used to create an example of one-dimensional Gorenstein monomial curves and we proved that the there is the relation for simple simple gluing the Betty number relation for simple gluing in case of affine monomial curve is given by Stamarte and we proved that the same relation hold for the projective closer also and we proved that if if original if gamma is a C gamma closure is arithmeticly Cohen Macaulay and gamma 1 is a simple gluing then the C gamma 1 closure is also arithmeticly Cohen Macaulay and Gorenstein if if original one is Gorenstein Gorenstein. So, that means the for affine case in case of a simple gluing the affine case and the result holds for the affine case as well as for the projective closer next is the simplicial affine semi group. So, Om Prakash defined the affine semi groups and I will define the simplicial affine say what is simplicial affine semi group. So, if let S be an affine semi group in A and D and affine semi group is said to be simplicial if there exists D linearly independent elements in the minimal generating set of S and for any element of semi group then the multiple of that elements can be written as the linear combination of those D elements over the natural number and we assume the as a simplicial affine semi group fully embedded in and D which means that it is subalgebra of the polynomial ring. Here R is R is a semi group ring corresponding to the affine semi group S and I s is the defining ideal of K s which is the kernel of the homomorphism phi which is defined by z i goes to x to the power a i and the associated grid ring of R is defined as this. So, for if you take D equals to n 1 that means the for numerical semi group the Sahin and Arsalan gave the Grobner basis criteria for checking the Cohen Macaulayness of the associated grid ring and for projective closer the associated grid ring is isomorphic to the semi group ring itself. So, and we extend the reserve for associated grid ring of a simplicial affine semi group ring and we prove that if if if there is a reduction ideal of the maximal ideal of semi group ring and semi group ring is Cohen Macaulay and G is a minimal Grobner basis of the defining ideal with respect to some special monomial ordering then associated grid ring of R is Cohen Macaulay if the variable corresponding to the those D elements do not divide the leading monomial of all f i's then we can say the associated grid ring is a Cohen Macaulay and we are also able to prove that the under some condition the betty numbers of the associated grid ring is same as the betty number of the semi group ring itself.