## Periodic Reporting for period 1 - Frobenius (Frobenius related invariants and singularities)

Reporting period: 2017-02-15 to 2019-02-14

"This is a project in commutative algebra, which has also many connections with algebraic geometry.

In fact, commutative algebra can be seen also as the algebraic approach to geometrical problems, that sometimes can be better solved in algebraic terms.

In this project, a particular emphasis is given to problems related to positive characteristic, that is the study of rings, and the corresponding varieties, over fields of prime characteristic.

Positive characteristic methods are difficult and require new approaches, quite different to the ones in characteristic zero. On the other hand, one often can use powerful techniques based on the Frobenius homomorphism, and therefore only available in positive characteristic, to get better results that afterwards can be lifted to characteristic zero.

There are three main mathematical objects of investigation in this project:

1) the F-signature function;

2) the differential symmetric signature;

3) the Veronese compactification V_{d,n}.

1) The F-signature is a numerical function which can be defined only for rings of positive characteristic by looking at the asymptotic splitting properties of the Frobenius homomorphism.

It has been introduced in 2004 by Huneke and Leuschke, who continued previous ideas of Smith and Van den Bergh.

The focus so far has been mainly on the leading term of the function, simply called F-signature, which already encodes a significant amount of information about the ring and its singularities.

Recently, Polstra and Tucker raised the question of whether a ""second coefficient"" for the function exists as well, i.e. whether the function can be written as

f(x)=a*x^d+b*x^{d-1}+O(x^{d-2}),

where a and b are real numbers. In this project, we investigate this question for specific classes of rings.

2) The differential symmetric signature is a new numerical invariant defined by the Experienced Researcher in his Ph.D. thesis in the attempt to find a characteristic-free version of the F-signature.

The differential symmetric signature proved to be equal to the F-signature for two-dimensional Kleinian singularities and cones over elliptic curves.

However, these examples are limited to the two-dimensional situation. In this project we investigate the differential symmetric signature for higher dimensional examples.

3) The Veronese compactification V_{d,n} is an algebraic projective variety which can be seen as the closure in the Zariski topology of the configuration space of n points in d-dimensional projective space lying on a common rational normal curve. The Veronese compactification is defined in every characteristic and has appeared in several settings in the last years, in particular in connection with certain moduli spaces.

Recently, Speyer and Sturmfels raised the question of finding explicit equations that cut out V_{d,n} set-theoretically. In this project we investigate this question."

In fact, commutative algebra can be seen also as the algebraic approach to geometrical problems, that sometimes can be better solved in algebraic terms.

In this project, a particular emphasis is given to problems related to positive characteristic, that is the study of rings, and the corresponding varieties, over fields of prime characteristic.

Positive characteristic methods are difficult and require new approaches, quite different to the ones in characteristic zero. On the other hand, one often can use powerful techniques based on the Frobenius homomorphism, and therefore only available in positive characteristic, to get better results that afterwards can be lifted to characteristic zero.

There are three main mathematical objects of investigation in this project:

1) the F-signature function;

2) the differential symmetric signature;

3) the Veronese compactification V_{d,n}.

1) The F-signature is a numerical function which can be defined only for rings of positive characteristic by looking at the asymptotic splitting properties of the Frobenius homomorphism.

It has been introduced in 2004 by Huneke and Leuschke, who continued previous ideas of Smith and Van den Bergh.

The focus so far has been mainly on the leading term of the function, simply called F-signature, which already encodes a significant amount of information about the ring and its singularities.

Recently, Polstra and Tucker raised the question of whether a ""second coefficient"" for the function exists as well, i.e. whether the function can be written as

f(x)=a*x^d+b*x^{d-1}+O(x^{d-2}),

where a and b are real numbers. In this project, we investigate this question for specific classes of rings.

2) The differential symmetric signature is a new numerical invariant defined by the Experienced Researcher in his Ph.D. thesis in the attempt to find a characteristic-free version of the F-signature.

The differential symmetric signature proved to be equal to the F-signature for two-dimensional Kleinian singularities and cones over elliptic curves.

However, these examples are limited to the two-dimensional situation. In this project we investigate the differential symmetric signature for higher dimensional examples.

3) The Veronese compactification V_{d,n} is an algebraic projective variety which can be seen as the closure in the Zariski topology of the configuration space of n points in d-dimensional projective space lying on a common rational normal curve. The Veronese compactification is defined in every characteristic and has appeared in several settings in the last years, in particular in connection with certain moduli spaces.

Recently, Speyer and Sturmfels raised the question of finding explicit equations that cut out V_{d,n} set-theoretically. In this project we investigate this question."

"Several activities were developed jointly by the Experienced Researcher and the Supervisor at the host institution.

These were aimed to enhance the transfer of knowledge required to achieve the goals mentioned in the previous section.

Here, we present a short summary of them. For a more detailed description and a complete list see Part B of the Final Report.

1) Seminar on Perfectoid Spaces

This was a weekly seminar whose main topic was Perfectoid Spaces, a class of algebro-geometric objects defined by the Field medalist Peter Scholze.

Despite their recent introduction, perfectoid spaces have already proved to be a powerful tool and have been used to tackle several open problems in different areas of Mathematics, including commutative algebra.

For example, Yves André used intensively perfectoid spaces in his proof of the Direct Summand Conjecture.

2) Seminar on Cohen-Macaulay representations

This was a weekly seminar, whose main topic was Cohen-Macaulay modules and their representation in low-dimension. The goal was to understand and master different techniques used to give a description of the category of Cohen-Macaulay modules over commutative rings of dimension smaller than 3.

3) Participation to international schools and conferences

During the execution of the project, the Experienced Researcher participated to several international schools and conferences in order to improve his knowledge on the hot research topics and disseminate the results obtained so far.

A complete list of the events attended and the talks given is included in Part B. Among them, here we just mention the workshop on ""The Homological Conjectures"" (March 2018, Berkeley, USA) and SIAM Conference on Applied Algebraic Geometry (July 2017, Atlanta, USA).

4) FACARD Workshop

As a further way to promote the results of the project, the Supervisor and the Experienced Researcher are organizing a workshop entitled ""FACARD. Frobenius Action in Commutative Algebra: Recent Developments"".

At the time of writing the workshop is planned for January 16-18, 2019 at the host institution.

It will consist of two minicourses at graduate level and research talks given by international experts and young emerging scientists."

These were aimed to enhance the transfer of knowledge required to achieve the goals mentioned in the previous section.

Here, we present a short summary of them. For a more detailed description and a complete list see Part B of the Final Report.

1) Seminar on Perfectoid Spaces

This was a weekly seminar whose main topic was Perfectoid Spaces, a class of algebro-geometric objects defined by the Field medalist Peter Scholze.

Despite their recent introduction, perfectoid spaces have already proved to be a powerful tool and have been used to tackle several open problems in different areas of Mathematics, including commutative algebra.

For example, Yves André used intensively perfectoid spaces in his proof of the Direct Summand Conjecture.

2) Seminar on Cohen-Macaulay representations

This was a weekly seminar, whose main topic was Cohen-Macaulay modules and their representation in low-dimension. The goal was to understand and master different techniques used to give a description of the category of Cohen-Macaulay modules over commutative rings of dimension smaller than 3.

3) Participation to international schools and conferences

During the execution of the project, the Experienced Researcher participated to several international schools and conferences in order to improve his knowledge on the hot research topics and disseminate the results obtained so far.

A complete list of the events attended and the talks given is included in Part B. Among them, here we just mention the workshop on ""The Homological Conjectures"" (March 2018, Berkeley, USA) and SIAM Conference on Applied Algebraic Geometry (July 2017, Atlanta, USA).

4) FACARD Workshop

As a further way to promote the results of the project, the Supervisor and the Experienced Researcher are organizing a workshop entitled ""FACARD. Frobenius Action in Commutative Algebra: Recent Developments"".

At the time of writing the workshop is planned for January 16-18, 2019 at the host institution.

It will consist of two minicourses at graduate level and research talks given by international experts and young emerging scientists."

All the scientific output is contained in peer-reviewed articles mentioned in the specific section.

A more detailed description of the results can be found in Part B.

1) De Stefani and I proved that the second coefficient of the F-signature function exists for invariant rings under the action of a finite small group whose order is not divided by the characteristic of the field.

Moreover, using notions from representation theory we are able to prove that in this setting the F-signature function takes the shape of a quasi-polynomial.

We are also able to give a description of the other coefficients in terms of invariants of the finite acting group.

When the group is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples of interest such as Veronese rings and Iyama-Yoshino’s singularities.

2) Brenner and I generalize our previous results on the differential symmetric signature to higher dimension.

We prove that the differential symmetric signature for invariant rings under the action of a finite small group G whose order is not divided by the characteristic of the field is equal to the value 1/|G|, which coincides with the F-signature.

We compute the differential symmetric signature for hypersurface rings of dimension ≥3 with an isolated singularity, obtaining the value 0.

While the first result extends a result of Watanabe and Yoshida from F-signature to the setting of differential symmetric signature, there is no analogue of the second one for F-signature.

Actually, we use it to exhibit an example of a ring where the differential symmetric signature and the F-signature are different.

3) Using the Gale transform and some inductive argument, Giansiracusa, Moon, Schaffler, and I construct polynomials whose associated variety W_{d,n} contains the Veronese compactification V_{d,n}.

Moreover, we prove that if d=2 or (d,n)=(3,7), (3,8), or (4,8) then W_{d,n}= V_{d,n}. In particular, in these cases we have equations that cut out V_{d,n}.

In addition, for n=d+4 or d=3 we are able to prove that W_{d,n} is given by the union of V_{d,n} and the determinantal variety that parametrizes configuration of points lying on a common hyperplane.

We also pinpoint several challenges involved in eliminating this extra component.

A more detailed description of the results can be found in Part B.

1) De Stefani and I proved that the second coefficient of the F-signature function exists for invariant rings under the action of a finite small group whose order is not divided by the characteristic of the field.

Moreover, using notions from representation theory we are able to prove that in this setting the F-signature function takes the shape of a quasi-polynomial.

We are also able to give a description of the other coefficients in terms of invariants of the finite acting group.

When the group is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples of interest such as Veronese rings and Iyama-Yoshino’s singularities.

2) Brenner and I generalize our previous results on the differential symmetric signature to higher dimension.

We prove that the differential symmetric signature for invariant rings under the action of a finite small group G whose order is not divided by the characteristic of the field is equal to the value 1/|G|, which coincides with the F-signature.

We compute the differential symmetric signature for hypersurface rings of dimension ≥3 with an isolated singularity, obtaining the value 0.

While the first result extends a result of Watanabe and Yoshida from F-signature to the setting of differential symmetric signature, there is no analogue of the second one for F-signature.

Actually, we use it to exhibit an example of a ring where the differential symmetric signature and the F-signature are different.

3) Using the Gale transform and some inductive argument, Giansiracusa, Moon, Schaffler, and I construct polynomials whose associated variety W_{d,n} contains the Veronese compactification V_{d,n}.

Moreover, we prove that if d=2 or (d,n)=(3,7), (3,8), or (4,8) then W_{d,n}= V_{d,n}. In particular, in these cases we have equations that cut out V_{d,n}.

In addition, for n=d+4 or d=3 we are able to prove that W_{d,n} is given by the union of V_{d,n} and the determinantal variety that parametrizes configuration of points lying on a common hyperplane.

We also pinpoint several challenges involved in eliminating this extra component.