 to work with Professor Mateo Barbaro. So here all the rings we assume are Noetherian and a prime characteristic. And the advantage of prime characteristic is we have Frobenius map. We know the definition that it's ring homomorphism which sends its elements to its pth power. And Frobenius plays an important role to study singularities. For instance, a ring is reduced if and only if the Frobenius is injective. Next we see what Frobenius does on polynomial ring. If r is a polynomial ring on fp, where fp is a field of p elements, then the r is a free module over fr with basis x1 to the power i1 dot, dot, dot, xn to the power in, where each ij's are between 0 to p minus 1. More generally, it's a classical result due to Kuhn's, which says that the ring is regular if and only if the Frobenius is a flat map. Now we relax the flatness condition in various ways. We get various kinds of singularities. And altogether, they are called f-singularities. So in the morning, Professor Watanabe defined this tight closure and weakly if regular and if rational. So I'll only say the last two, which is a pure. A pure means that the Frobenius morphism is a pure map. And if injective, meaning that the Frobenius action on the local homology module is injective. Now this shows the relations between various f-singularities. Now the examples is summands of polynomial rings are f-regular. In particular, normal affine semi-group rings are f-regular. Now again, these examples appear in the lecture of Professor Watanabe, that r is a ring kxyz modulo xq plus yq plus zq, where p is greater than 3, the characteristic. Then r is a pure if and only if p is congruent to 1 more 3. But it's never a regular. So it's not a regular as shown by Professor Watanabe. And this example actually shows that the f-singularity really depends on the characteristic. And the implications are strict implication. Now we move on to the Grobner deformation part. So let's quickly recall what is the monomial order. Let S be a polynomial ring over a field k. A monomial order is a total order which satisfies the two properties that one is the least element. And the second one says that the order respects multiplication by monomials. Then we say what is the definition of initial ideal. Let i be an ideal and less than be a monomial order. Define the initial ideal of i as the initial ideal of f, the ideal generated by in f, where in f is the biggest monomial appearing in f with respect to that ordering. Given a monomial order, it is possible to find a weight ordering w, where in i is same as in w. Where in w, i is the same as in f generated by the ideal in f, where in f is again the sum of terms of f with maximal w degree. The advantage of this realization is that in w, i can be realized as a deformation. So let f be a polynomial with homogeneous component fj. And then we want to homogenize it with a, we put an extra variable t. And we multiply t to the power of some suitable power so that it becomes a homogeneous of degree of f. And we call that as homogenization of f. That is fh. And home i is the set of elements generated by fh, where fh is the, again, homogenization of f. Then we say that home i is a homogeneous ideal. It is clear. And with the ordering is just w1, w2, wn, comma 1, where t is given order 1. Then r is called st mod home wi is called a Grobner deformation, where r is a standard graded ring. t is given the order 1. And all the degrees of xi's are wi's. The t is a non-zero divisor. And then the r mod tr is the s mod initial ideal means it's a special fiber. And then r mod t minus 1r is same as s mod i. This is a general fiber. So the general question we ask that how to obtain properties from s mod initial ideal to s mod i. So let i be a homogeneous ideal of s and less than be a monomial order on s. Then it is known that the Hilbert functions of s mod i and s mod initial ideal are same. Hence, their dimensions are same. The depth of s mod i is greater or equal to depth of s mod initial ideal. Hence, if s mod initial ideal is coin macaulay, then s mod i is so. Again, this is the same true for Gorenstein case. And for the regularity, regularity of s mod i is less than equal to regularity of s mod initial ideal. A recent result of Konka and Barbaro says that that i and initial ideal are much more related, provided that initial ideal is a square free monomial ideal. Precisely, they prove that if i is a homogeneous ideal and in i is square free monomial ideal, then the depths are actually equal. That is that s mod i is coin macaulay if and only if s mod initial ideal is coin macaulay. And the regularities are also same. Actually, they prove a stronger version and this comes as a corollary of their results. So in view of these results, we ask the following two questions that let i be an ideal of a polynomial ring s over a field k when there is a monomial order less than such that initial ideal is square free. So there are some well-known examples are there. One is the binomial edge ideals. Another one is algebras with strengthening laws. So here we identify a new class. Then the second one is that for which kinds of f singularities do we have s mod i has those f singularities. Provided with respect to some weight ordering, s mod initial ideal has those f singularities. So this is the new class we obtain that s be a polynomial ring in n variable over a field k, not necessarily prime characteristic. Let i be a radical ideal and let h be the maximum of the height of all minimal primes. If in i, i symbolic power h contains a square free monomial, then in i is a square free monomial ideal. And the proof is actually long. The idea is we first prove it for prime characteristic, then derive it for characteristic zero. And then the prime characteristic proof relies on a version of Federer's criteria. Then we go to the second questions. So we give negative answers for the questions that s mod initial ideal have f regularity, but s mod i does not have. So this is the example taken from Anurag Singh's work. So s be a polynomial ring in five variables that i be the ideal generated by two by two minors. Then if we consider the weight six, 24, six, three, one, then initial ideal is the ideal of this matrix where x to the power five is not there. Then again by a work of Anurag Singh, s mod initial ideal is f regular, but s mod i is not f regular. The negative answer for the f purity, the same example, but now the characteristic is greater than three. Then we take the ordering is now changed. We take the lexicographic ordering x one greater than x two greater than x three greater than x three and x four. Then the initial ideal is a square free monomial ideal. And hence it is a pure, but again the same for p greater than three it is not even a pure. It's a by work of Anurag Singh. So now there is another singularity which fits in between a purity and f injectivity. It is strongly f injectivity. So I'll not define this here. This is as f injective plus f full. And we give positive answers for f rationality and strong f injectivity. Then we say that let s be a polynomial ring over prime characteristic, w be a weight ordering. If I be an ideal such that s mod initial ideal has a rational or respectively strongly f injective then s mod I has that kind of singularities. So one important corollary is that if s be a polynomial ring or prime characteristic and less than be a monomial order such that initial ideal is radical then s mod I is always strongly f injective. In particular this is f injective. And this example gives various f singularities of various kinds of algebras. First one is the binomial edge algebras. So given any graph we can define a binomial ideal which is called binomial edge algebras. And it is known that there exists a ordering shows that the initial ideal is square free. Hence as a corollary we get that the binomial edge algebras is always f injective. Then the second one is the f singularities of algebras with straightening law. So the definition is big. So how much time I have? Five minutes? Okay, so maybe I'll give the definition. So let a be a n graded algebra and h be a finite poset. And suppose there is an injective map from h to union of ai. The elements of h will be identified with their images. Given a chain h1 less than equal to h2 less than equal to hs of elements. We correspond to the product h1 to hs called the standard monomial. And one say that a is algebras with straightening laws or asl with respect to h if three conditions satisfies. The first condition is the element of h generate a as an a not algebra. Second one is the standard monomials are a not linearly independent. And then the third one is the straightening law relation that even h1 and h2 which are not comparable then there exist a relation which says that h1 and h2 h1.h2 is equal to the summation of standard monomials where we have this that hj1 the first term is strictly less than h1 and it is also strictly less than h2. So the classical examples of algebras with straightening laws are determinant ideals various determinant ideals. The first one is just the take a matrix of two by two matrix x1, x2, x1, x1, 1, x1, 2, x2, 1, x2, 2 and then consider the determinant of this matrix is this. And then r is a asl with this poset where x1 one to an x2 one are not comparable others are comparable. And again one can show that there exist a reverse lexicographic ordering refining the ordering on the poset that any asl has a square free initial ideal. Hence it is always strongly f-injective. Now we conclude with this the following example. So let S be a polynomial ring in four variables then K is algebraically closed P greater than zero and I be the ideal of this two by two minor. And by a work of Hoxter and Huneke that S mod I is not a pure. The point is that this example can be given as a asl structure with this that where four variables x1, x2, x3 and x4 where x4 is less than x1, x2, x3 and all x1, x2, x3 are not comparable. And then this is by the terminology of Eisenberg this is an wonderful poset and an an conjecture of Eisenberg states that any asl on a wonderful poset is a pure. This gives the counter example of that conjecture because this is not a pure and this poset is wonderful. So thank you for listening my talk here. So this is the references. Thank you.