 Good afternoon. Excuse me, my English is poor. So try to speak as best as possible. In this opportunity, I will talk about on the dimension, and many dimension of the formal versus space of the model is a curve, monomial curves, and the linear formula. First, remember, what is a bias of point? Let's see a complete irreplaceable smooth curve of genus G and p-closet point in C. Consider the following. As in the chain vector space, where vector space of the set of rational function has a pole of the order of most n in p? Is it equality? Is it valid? Is n is called a gap? My theorem primer log quantity of gaps is equal to G. So the semi-global associated of the point is denoted as p is equal to natural numbers, minus of the gaps. We say the p is an ordinal point if vector space is 0, when n is equal to G. Another case is bias of point. That is sp different than that set. MGL admits a certification with semi-group associated the point of C, where any two semi-groups is prime with half intersection is empty. A natural question is, what is the dimension of the model with space fixed at s? And as a question, we help describe show ring to space MGL. There are two results nested in this direction. A field, a formula, delineate formula, an upper bound. This is a formula delineate, where n sets in the morphine of s. And second result is lower bound, given the native pluger, a student of the ISMAD. That's improving a lower bound. It's called a 3G and 2 effective weight of s. The effective weight of s is a sum of quantity of generators as of s with a mean or l. And two result, push up. But there are families semi-groups where other stick inequality holds and samples later. Using the techniques on the control store to compute the upper bound on the dimension as symmetric groups. Symmetric means in the bracket, GAPI is equal to 2G minus 1. As we call, it's in that conjugate. This conjugate to psi dimension is delineate formula minus that quantity, where teum most is positive greater part of the cotangent complex teum t1 associated to the range of regular functions of the curve monomial affine. Follow the paper, lechinger, we can see on the description of teum is equal to cotangent of the sequence xz. This sequence associated with the monomial curve. A result is psi if s n over 4, then a dimension of the teum graded in s is 0, or a dimension is this formula. When s is a set of the e, l is equal to the quantity of the generators of the semi-group s. And v s, l is a vector space generator gamma jot. Our key and b2d property does if s is negative graded, that means a dimension in graded n is equal to 0. For any n natural, then a dimension is equal of the formula, or delineated formula. Another result is in the paper, written vitally, they have a description of the negative graded semi-group. Nathan showed that s is negative graded. Then, as of the formulas dimension, lower and upper are the sum is equal. With this addition? Negative group. No, in Spanish. Can you give me an example? Do you know any examples of negative graded n? Yes. In fact, there is an example. s is a generator of 6, 0 to 8, 9, 10. This curve, as I said, the semi-group is negative graded. It's a semi. Generally, I think that k6, a1, my 6, 0, a2, my 6, 10. The same plus the semi-group, negative graded. The first generator is the benign. In Italian? For semi-group, this form, a sequence is valid. The following most showed that the dimension is comparative of the dimension of the native, the leader of the following gaps. For example, this gap is a key here. The dimension of the native is 9. The leader is 10, but a conjecture, say, the leader minus a dimension of the T most. T here, we have a quality, 9, 9. In general, a dimension of the leader is most comparative of the dimension of data. As it is, an exercise knows the difficulties. It's valid here. Minus a dimension T most. There exists a semi-group S, a dimension of the model space is 30. It's easy to see that formula here 14 in the sidelight. Using the techniques of the container store and Prüger, Loverbond, we have three dimensions here. For example, if S is generated for five elements, where 6 here is a multiplicity of the curve. In this case, a dimension is a tau must 7. If S is the form, a dimension is the form. For each one in the tree of all layers, some have a quality of the dimension. In this case, a dimension here is 0. The question is that the semi-group in general is true that is valid or followed. A dimension is an equation linear in tau. The key is S when tau is equal to 1 is a question. The linear formula, we really learn a lot the linear formula problem, but let's see project Aljebra curve, the final of the key and a point, a closer point, the feeder. Following notation, all denotes local ring in the point and all tube is normalization. A dimension of the quotient is a singularity degree in the point. There O is a model of derivations of O. Mu 1 is a dimension of the quotient, minus a dimension of this quotient. The linear formula says that the irreversible component of the formalversal deformation of the speck of O. If the fiber above the generic point of E is smooth, then a dimension is delta mu. Approve is not depends on the completations. Then we use a symbol for the local ring O and its completations O. And we saw the singular point of C. And so the universal deformation of C, those exits in this, we have a diagram commutative. And they're not there on the only closer point of T by the semi-universal property. The universal deformation of C is giving morphine T0 and T and it's morphine product space. From the terminal ring side, this may be smooth. It is flat with smooth fiber. By the dimension of universal property, any fields of the deformation of C, say it's number duality, there is an ambidextion in the morphine and morphine and hence the element in the tangent space T0, T0. Sine-K is the only singular point of C at fields of the deformation of C is locally trivial. And only if the endocid field deformation is trivial is equivalent to the composite form morphine. Define a zero vector in the tangent space theta T at T. Thus, we had a rejection between locally trivial fields of the deformation of C and zero vector equivalent to KfNl of the map. By collider, Spencer correspondents obtain an ambidextion here. Well, T C said that the derivation is tangent chief of C. The thing, by him, B2D, T is smooth and then following the sequence and hence a dimension is of the fiber theory. We have this formula here which is equality here. Let E be a component irreplaceable of the basal space of the formal version deformation of the E into the key inverse pre-image of the underpin. Then here equal dimension of E is zero minus dimension key. Well, you put assume that a formal version of the formation of the spec is smooth above a general point of E. It is the image across the sub-scheme of X given by the geocon Jacob ideal does not contain the general point E. Later, as concrete count, other dimension of the E is 3G minus 3 of juice characteristic formula. Let's see a non-singular model of C then obtain a difference where the graph singularity is locally free of range 1. The degree of T is given for this and hence is unformally. I assume that the degree of T is negative here. The result says we have a follow result is valid equality and we talk as I talk for family semi-group of C is positive a conjecture the idea is used that with space model with fixed S is in projectivation of T and a bond for its dimension bibliography reference.