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This is a survey paper on algebraic surfaces in positive characteristic based on a series of lectures that the author gave at the University of Edinburgh in March 2023. It is focused on certain positive characteristic phenomena like infinitesimal group schemes and their actions on algebraic surfaces as well as the failure in positive characteristic of certain fundamental characteristic zero results like the Kodaira vanishing theorem. Many explicit examples are presented., Comment: The paper is for the final Edge volume in the ADUF journal

Soil Organic Carbon (SOC) constitutes a fundamental component of terrestrial ecosystem functionality, playing a pivotal role in nutrient cycling, hydrological balance, and erosion mitigation. Precise mapping of SOC distribution is imperative for the quantification of ecosystem services, notably carbon sequestration and soil fertility enhancement. Digital soil mapping (DSM) leverages statistical models and advanced technologies, including machine learning (ML), to accurately map soil properties, such as SOC, utilizing diverse data sources like satellite imagery, topography, remote sensing indices, and climate series. Within the domain of ML, self-supervised learning (SSL), which exploits unlabeled data, has gained prominence in recent years. This study introduces a novel approach that aims to learn the geographical link between multimodal features via self-supervised contrastive learning, employing pretrained Vision Transformers (ViT) for image inputs and Transformers for climate data, before fine-tuning the model with ground reference samples. The proposed approach has undergone rigorous testing on two distinct large-scale datasets, with results indicating its superiority over traditional supervised learning models, which depends solely on labeled data. Furthermore, through the utilization of various evaluation metrics (e.g., RMSE, MAE, CCC, etc.), the proposed model exhibits higher accuracy when compared to other conventional ML algorithms like random forest and gradient boosting. This model is a robust tool for predicting SOC and contributes to the advancement of DSM techniques, thereby facilitating land management and decision-making processes based on accurate information., Comment: Accepted for publication in IEEE Transactions on Geoscience and Remote Sensing (TGRS)

We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristics. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, which is a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with is twisted forms. The result actually holds true for arbitrary proper integral schemes under the assumption that the Frobenius kernel has large isotropy group at the generic point. This property is measured by a new numerical invariant called the foliation rank., Comment: 42 pages, bounds in Theorem 13.2 improved, discussion of Raynaud surfaces and Tango curves in Section 14 included, otherwise minor changes, to appear in Algebra Number Theory

This paper studies the existence of non trivial $\mu_p$ actions on a canonically polarized surface X defined over an algebraically closed field of characteristic p>0. In particular, an explicit function $f(K_X^2)$ is obtained such that if $p>f(K_X^2)$, then there does not exist a non trivial $\mu_p$-action on X. This implies that the connected component of the automorphism scheme of X containing the identity is either smooth or is obtained by successive extensions by $\alpha_p$., Comment: 45 pages

This paper investigates the geometry of canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces. In particular, an explicit integer valued function f(x) is obtained with the following properties. If X is a canonically polarized surface with canonical singularities defined over an algebraically closed field of characteristic p>0 such that p>f(K_X^2) and X has a nontrivial global vector field, then X is unirational and the algebraic fundamental group is trivial. As a consequence of this result, large classes of canonically polarized surfaces are identified whose moduli stack is Deligne-Mumford, a property that does not hold in general in positive characteristic. This paper is mathematically identical to the previous version. The reason that the paper is replaced is in order to point out that this paper is a generalization to the case of singular surfaces with canonical singularities of the paper "Vector fields and moduli of canonically polarized surfaces in positive characteristic" with reference arXiv:1710.03076 which treated only the case of smooth surfaces. The results of this paper supercede the results of the aforementioned paper making it obsolete., Comment: This paper is a generalization to the case of singular surfaces with canonical singularities of the paper "Vector fields and moduli of canonically polarized surfaces in positive characteristic" with reference arXiv:1710.03076 which treated only the case of smooth surfaces. The results of this paper supercede the results of the aforementioned paper making it obsolete. 46 pages

This paper investigates the geometry of smooth canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces. In particular, an explicit real valued function f(x) is obtained such that if $X$ is a smooth canonically polarized surface defined over an algebraically closed field of characteristic p>0 such that $K_X^2

This paper investigates the geometry of a smooth canonically polarized surface $X$ defined over an algebraically closed field of characteristic $p>0$ in the case when the automorphism scheme of $X$ is not smooth. This is a situation that appears only in positive characteristic and it is closely related to the structure of the moduli stack of canonically polarized surfaces. Restrictions on certain numerical invariants of $X$ are obtained in order for Aut(X) to be smooth or not and information is provided about the structure of the component of Aut(X) containing the identity. In particular, it is shown that a smooth canonically polarized surface X with 0< K_X^2 < 3 and non smooth automorphism scheme tends to be uniruled and simply connected. Moreover, X is the purely inseparable quotient of a ruled or rational surface by a rational vector field., Comment: This paper is a generalization to all characteristics of the paper "Automorphisms of smooth canonically polarized surfaces in characteristic 2", arXiv:1408.1291, of the same author. The results of this paper completely supersede the results of the aforementioned paper making it obsolete. 64 pages

This paper studies schemes X defined over a field of characteristic p>0 which admit a nontrivial $\alpha_p$ or $\mu_p$ action. In particular, the structure of the quotient map $X \rightarrow Y$ is investigated. Information on local properties of the quotient Y, as singularities and local Picard groups, structure theorems for the quotient map and adjunction formulas for the quotient map are obtained., Comment: Revised version. A section has been added which contains additional material about the singularities of the quotient of a normal scheme by a nontrivial $\mu_p$ or $\alpha_p$ action. 34 pages

This paper investigates the structure of the automorphism scheme of a smooth canonically polarized surface $X$ defined over an algebraically closed field of characteristic 2. In particular it is investigated when Aut(X) is not smooth. This is a situation that appears only in positive characteristic and it is closely related to the structure of the moduli stack of canonically polarized surfaces. Restrictions on certain numerical invariants of X are obtained in order for Aut(X) to be smooth or not and information is provided about the structure of the component of Aut(X) containing the identity. In particular, it is shown that if X is a smooth canonically polarized surface with 1\leq K_X^2 \leq 2 with non smooth automorphism scheme, then X is uniruled. Moreover, if K_X^2=1, then X is simply connected, unirational and p_g(X) \leq 1. Moreover, X is the purely inseparable quotient of a rational surface by a rational vector field., Comment: The main theorem has been strengthened compared to the last version of the paper. It now includes surfaces with K^2=2

We describe the sheaf T^1(X) of first order deformations of a reduced scheme with normal crossing singularities. In particular, we obtain a formula for T^1(X) in a suitable log resolution of X., Comment: 16 pages

This paper obtains criteria for a Fano variety X with normal crossing singularities defined over an algebraically closed field of characteristic zero, to be smoothable. The difference with the original version is that the theory of logarithmic structures and deformations is used in order to prove that X is smoothable by a smooth variety, if and only if T^1(X)=O_D, where D is the singular locus of X., Comment: 19 pages. Final version. To appear in the Proceedings of the Edinburgh Mathematical Society

We study the deformation theory of a Fano variety X with normal crossing singularities of dimension at most three. We obtain a formula for the sheaf T^1(X) of first order deformations of X in a suitable log resolution of X and its singular locus Z and we obtain explicit criteria for the existence of smoothings of X., Comment: 22 pages

In this paper we study the deformation and Q-Gorenstein deformation theory of schemes with non-isolated singularities. We obtain obstruction spaces for the existence of deformations and also for local deformations to exist globally. Finally we obtain explicit criteria in order for a pure and reduced scheme of finite type over a field $k$ to have smoothings and Q-Gorenstein smoothings., Comment: Final version, accepted for publication in the Michigan Mathematical Journal. 53 pages

Let X be the germ of a Gorenstein 3-fold singularity and C a smooth curve through it such that the general hyperplane section S of X containing D is DuVal of type E6 or E7. In this paper we obtain criteria for the existence of a terminal divisorial extremal neighborhood f: Y-->X contracting an irreducible divisor E onto C and classify all such neighborhoods., Comment: A section has been added classifying all divisorial extremal neighborhoods f:Y-->X such that if E is the f-exceptional divisor and C=f(E), the general hyperplane section S of X containg C is DuVal of type E6. The previous version classified only the E7 case. 20 pages

Revised Version. An example of a locally smoothable stable surface that does not have a global smoothing has been added., Comment: 27 pages

Revised version. Proper credits added. In this paper we study divisorial extremal neighborhoods f:Y-->X, such that X is a cAn type three dimensional terminal singularity, and C=f(E) is a smooth curve, where E is the f-exceptional divisor. We view a divisorial extremal neighborhood as a one-parameter smoothing of certain surface singularities, and based on this we give a classification of such neighborhoods., Comment: 37 pages

Let X/T be a one parameter family of canonical 3-folds and let D be a Weil divisor on it flat over T. We study the problem of when the D_t-minimal models of X_t form a family and we obtain conditions for this to happen. As an application of this we classify terminal divisorial contractions Y-->X that contract an irreducible divisor E onto a smooth curve C in the case when the general hyperplane section S of X through C is a D5 DuVal singularity., Comment: Revised version. A gap in the proofs of Lemma 2.3 and Theorem 2.4 is corrected. 26 pages

In order to calculate the multiplicity of an isolated rational curve C in a local complete intersection variety X, i.e. the length of the Hilbert scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X. This is equivalent to infinitesimal extensions of P^1 by locally free sheaves. In this paper we study infinitesimal extensions of P^1, determine their structure and obtain upper and lower bounds for the length of the local rings of their Hilbert schemes at [P^]., Comment: 22 pages

Let C be a smooth curve on an index 1 terminal 3-fold. We investigate the existence of extremal terminal divisorial contractions Y-->X that contract an irreducible surface E to C. We consider cases in respect to the singularities of the general hypersurface section S of X through C. We completely classify such contractions in the case that S has A_i, i < 4 and D_2n for any n, type singularities., Comment: 32 pages