## Final Report Summary - MODPOSCHAR (Moduli of canonically polarized varieties in positive characteristic)

OBJECTIVES OF THE PROJECT.

The objective of this research project was the investigation of the moduli of canonically polarized varieties defined over an algebraically closed field of positive characteristic. In particular it aimed to investigate the following general problem.

Problem: What is the structure and the basic properties of the moduli space and moduli stack of stable varieties with fixed Hilbert polynomial defined over an algebraically closed field of characteristic p>0? In particular, are they proper or of finite type? Is the stack Deligne-Mumford? In the case when one of these properties fails, why does it fail and how can the moduli problem be modified in order to satisfy them?

This problem is within the context of classification of varieties of general type, a fundamental problem of algebraic geometry. In characteristic zero this problem has been extensively studied. However, before this project very little was known in positive characteristic.

WORK PERFORMED.

The research conducted so far has been limited to the two-dimensional case of the problem. This is the first nontrivial case and it is the basis for the understanding of the general case.

In particular the following two problems have been investigated:

1. What is the geometry of a smooth canonically polarized surface with non-reduced automorphism scheme, or equivalently with nontrivial global vector fields? Are there numerical relations between the characteristic of the base field and numerical invariants of the surface which imply the smoothness of the automorphism scheme?

2. What is the structure of the quotient of a smooth surface by a nontrivial action of the additive or multiplicative infinitesimal group scheme?

The first problem is intimately related to the structure of the moduli stack of canonically polarized surfaces, one of the main objectives of this project, for the reasons that will be explained in the next paragraph. The second problem provides some of the necessary technical machinery needed in order to study the first problem.

Recent results show that the moduli stack of canonically polarized surfaces is of finite type over the base field. However, unlike the characteristic zero case, it is not Deligne-Mumford. This is an unpleasant failure because the property that the stack is Deligne-Mumford implies the existence of a family universal in the etale topology. The reason that the stack is not Deligne-Mumford is that in positive characteristic there exist smooth canonically polarized surfaces with non-reduced automorphism scheme, or equivalently with nontrivial global vector fields. This is a situation that appears exclusively in positive characteristic. Therefore the non-reducedness of the automorphism scheme is the obstruction for the moduli stack to be Deligne-Mumford.

MAIN RESULTS OBTAINED.

The research conducted within this project has been in two phases.

During the first phase the structure of the quotient Y of a smooth surface X by a nontrivial action of the additive or multiplicative infinitesimal group scheme has been studied. Information about the singularities of the quotient Y as well as structure theorems and adjunction formulas for the quotient map X—>Y have been obtained. In particular, it has been shown that in the case of the quotient by the action of the multiplicative group scheme, the quotient map is a torsor in an open neighbourhood of any connected component of the fixed locus of the action.

During the second phase the geometry of a smooth canonically polarized surface with non-reduced automorphism scheme has been investigated. This was the core of the research conducted so far. The main results are the following:

Theorem 1. Let X be a smooth canonically polarized surface which lifts to characteristic zero or at least to the second order Witt vectors. Then the automorphism scheme of X is reduced.

This result immediately implies that smooth hypersurfaces or complete intersections in projective space have reduced automorphism scheme.

Theorem 2. Let f(x) be a polynomial with rational coefficients. Then there are positive integers m and M depending only on f(x) such that for any Gorenstein canonically polarized surface X with at worst canonical singularities defined over an algebraically closed field k of characteristic p>0 and with Hilbert polynomial f(x), the length of the automorphism scheme Aut(X) of X is at most M. Moreover, Aut(X) is reduced if p>m.

This result shows that non-reducedness of the automorphism scheme happens for relatively small values of the characteristic of the base field compared to certain numerical invariants of the surface. Consequently, the moduli stack is Deligne-Mumford if the characteristic of the base field is large enough.

Theorem 3. Let X be a smooth canonically polarized surface defined over an algebraically closed field k of characteristic p>0 such that the automorphism scheme of X is not reduced and its first Chern class is at most 2. Then:

1. Suppose that the first Chern class of X is 2 and that p is not 3 or 5. Then X is uniruled.

2. Suppose that the first Chern class of X is 1 and that p is not 7. Then X is unirational and simply connected, except possibly if p is 3 or 5 and X is a simply connected supersingular Godeaux surface. Moreover, if p is not 3,5 then its geometric genus is at most 1.

In all cases, X is the quotient of a ruled surface by a rational vector field.

This result shows that a canonically polarized surface with non-reduced automorphism and small invariants tends to be unruled, a property that is exclusive in positive characteristic. Moreover the result shows that non-reducedness of the automorphism scheme imposes strong restrictions on certain numerical invariants of the surface like its algebraic fundamental group and its euler characteristic. From this it follows that the moduli stacks of canonically polarised surfaces with certain numerical invariants are Deligne-Mumford. In particular the moduli stack of surfaces with first chern class 1 and nontrivial fundamental group or euler characteristic 3 is Deligne-Mumford.

PUBLICATIONS DERIVED FROM THE PROJECT.

1. Automorphisms of smooth canonically polarized surfaces in positive characteristic, To appear in Advances in Mathematics, DOI 10.1016/j.aim.2017.02.002, 56 pages.

2. Quotients of schemes by $\alpha_p$ or $\mu_p$ actions in characteristic p>0, Manuscripta Mathematica (2017), 152, 247-279.33 pages.

The objective of this research project was the investigation of the moduli of canonically polarized varieties defined over an algebraically closed field of positive characteristic. In particular it aimed to investigate the following general problem.

Problem: What is the structure and the basic properties of the moduli space and moduli stack of stable varieties with fixed Hilbert polynomial defined over an algebraically closed field of characteristic p>0? In particular, are they proper or of finite type? Is the stack Deligne-Mumford? In the case when one of these properties fails, why does it fail and how can the moduli problem be modified in order to satisfy them?

This problem is within the context of classification of varieties of general type, a fundamental problem of algebraic geometry. In characteristic zero this problem has been extensively studied. However, before this project very little was known in positive characteristic.

WORK PERFORMED.

The research conducted so far has been limited to the two-dimensional case of the problem. This is the first nontrivial case and it is the basis for the understanding of the general case.

In particular the following two problems have been investigated:

1. What is the geometry of a smooth canonically polarized surface with non-reduced automorphism scheme, or equivalently with nontrivial global vector fields? Are there numerical relations between the characteristic of the base field and numerical invariants of the surface which imply the smoothness of the automorphism scheme?

2. What is the structure of the quotient of a smooth surface by a nontrivial action of the additive or multiplicative infinitesimal group scheme?

The first problem is intimately related to the structure of the moduli stack of canonically polarized surfaces, one of the main objectives of this project, for the reasons that will be explained in the next paragraph. The second problem provides some of the necessary technical machinery needed in order to study the first problem.

Recent results show that the moduli stack of canonically polarized surfaces is of finite type over the base field. However, unlike the characteristic zero case, it is not Deligne-Mumford. This is an unpleasant failure because the property that the stack is Deligne-Mumford implies the existence of a family universal in the etale topology. The reason that the stack is not Deligne-Mumford is that in positive characteristic there exist smooth canonically polarized surfaces with non-reduced automorphism scheme, or equivalently with nontrivial global vector fields. This is a situation that appears exclusively in positive characteristic. Therefore the non-reducedness of the automorphism scheme is the obstruction for the moduli stack to be Deligne-Mumford.

MAIN RESULTS OBTAINED.

The research conducted within this project has been in two phases.

During the first phase the structure of the quotient Y of a smooth surface X by a nontrivial action of the additive or multiplicative infinitesimal group scheme has been studied. Information about the singularities of the quotient Y as well as structure theorems and adjunction formulas for the quotient map X—>Y have been obtained. In particular, it has been shown that in the case of the quotient by the action of the multiplicative group scheme, the quotient map is a torsor in an open neighbourhood of any connected component of the fixed locus of the action.

During the second phase the geometry of a smooth canonically polarized surface with non-reduced automorphism scheme has been investigated. This was the core of the research conducted so far. The main results are the following:

Theorem 1. Let X be a smooth canonically polarized surface which lifts to characteristic zero or at least to the second order Witt vectors. Then the automorphism scheme of X is reduced.

This result immediately implies that smooth hypersurfaces or complete intersections in projective space have reduced automorphism scheme.

Theorem 2. Let f(x) be a polynomial with rational coefficients. Then there are positive integers m and M depending only on f(x) such that for any Gorenstein canonically polarized surface X with at worst canonical singularities defined over an algebraically closed field k of characteristic p>0 and with Hilbert polynomial f(x), the length of the automorphism scheme Aut(X) of X is at most M. Moreover, Aut(X) is reduced if p>m.

This result shows that non-reducedness of the automorphism scheme happens for relatively small values of the characteristic of the base field compared to certain numerical invariants of the surface. Consequently, the moduli stack is Deligne-Mumford if the characteristic of the base field is large enough.

Theorem 3. Let X be a smooth canonically polarized surface defined over an algebraically closed field k of characteristic p>0 such that the automorphism scheme of X is not reduced and its first Chern class is at most 2. Then:

1. Suppose that the first Chern class of X is 2 and that p is not 3 or 5. Then X is uniruled.

2. Suppose that the first Chern class of X is 1 and that p is not 7. Then X is unirational and simply connected, except possibly if p is 3 or 5 and X is a simply connected supersingular Godeaux surface. Moreover, if p is not 3,5 then its geometric genus is at most 1.

In all cases, X is the quotient of a ruled surface by a rational vector field.

This result shows that a canonically polarized surface with non-reduced automorphism and small invariants tends to be unruled, a property that is exclusive in positive characteristic. Moreover the result shows that non-reducedness of the automorphism scheme imposes strong restrictions on certain numerical invariants of the surface like its algebraic fundamental group and its euler characteristic. From this it follows that the moduli stacks of canonically polarised surfaces with certain numerical invariants are Deligne-Mumford. In particular the moduli stack of surfaces with first chern class 1 and nontrivial fundamental group or euler characteristic 3 is Deligne-Mumford.

PUBLICATIONS DERIVED FROM THE PROJECT.

1. Automorphisms of smooth canonically polarized surfaces in positive characteristic, To appear in Advances in Mathematics, DOI 10.1016/j.aim.2017.02.002, 56 pages.

2. Quotients of schemes by $\alpha_p$ or $\mu_p$ actions in characteristic p>0, Manuscripta Mathematica (2017), 152, 247-279.33 pages.