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In this paper, a curve is any projective scheme of pure dimension one. It is well known that the arithmetic genus and the degree of an aCM curve D in P 3 are computed by the h-vector of D. However, for a given curve D in P 3 , the two aforementioned invariants of D do not tell us whether D is aCM or not. If D is an aCM curve on a smooth surface X in P 3 , any member of the linear system | D + l C | is also aCM for each non-negative integer l, where C is a hyperplane section of X. By a previous work, if a non-zero effective divisor D of degree d and arithmetic genus g on a smooth quintic surface X in P 3 is aCM and satisfies the condition h 0 (O X (D - C)) = 0 , then 0 ≤ d + 1 - g ≤ 4 . In this paper, we classify non-aCM effective divisors on smooth quintic surfaces in P 3 of degree d and arithmetic genus g such that 0 ≤ d + 1 - g ≤ 4 , and give several examples of them. [ABSTRACT FROM AUTHOR]

This paper is based on a series of talks that the author gave at the University of Edinburgh in March 2023 on surfaces in positive characteristic. In particular, infinitesimal group schemes and their actions on algebraic surfaces are discussed, the structure of the automorphism group scheme of a surface of general type and the failure of the Kodaira vanishing theorem. [ABSTRACT FROM AUTHOR]

In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying K 2 = 4 p g - 8 , for any even integer p g ≥ 4 . These surfaces also have unbounded irregularity q. We carry out our study by investigating the deformations of the canonical morphism φ : X → P N , where φ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of φ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of φ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of X are unobstructed even though H 2 (T X) does not vanish. Consequently, X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality p g > 2 q - 4 , with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with K 2 = 2 p g - 4 , studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus g ≥ 3 , for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map. [ABSTRACT FROM AUTHOR]

Surfaces of general type with canonical map of degree d bigger than 8 have bounded geometric genus and irregularity. In particular the irregularity is at most 2 if d ≥ 10 . In the present paper, the existence of surfaces with d = 10 and all possible irregularities, surfaces with d = 12 and irregularity 1 and 2, and surfaces with d = 14 and irregularity 0 and 1 is proven, by constructing these surfaces as Z 2 3 -covers of certain rational surfaces. These results together with the construction by C. Rito of a surface with d = 12 and irregularity 0 show that all the possibilities for the irregularity in the cases d = 10, d = 12 can occur, whilst the existence of a surface with d = 14 and irregularity 2 is still an open problem. [ABSTRACT FROM AUTHOR]

The purpose of this article is twofold. Firstly, we address and completely solve the following question: Let (X, L) be a smooth, hyperelliptic polarized variety and let φ : X ⟶ Y ⊂ P N be the morphism induced by |L|; when does φ deform to a birational map? Secondly, we introduce the notion of "generalized hyperelliptic varieties" and carry out a study of their deformations. Regarding the first topic, we settle the non trivial, open cases of (X, L) being Fano-K3 and of (X, L) having dimension m ≥ 2 , sectional genus g and L m = 2 g . This was not addressed by Fujita in his study of hyperelliptic polarized varieties and requires the introduction of new methods and techniques to handle it. In the Fano-K3 case, all deformations of (X, L) are again hyperelliptic except if Y is a hyperquadric. By contrast, in the L m = 2 g case, with one exception, a general deformation of φ is a finite birational morphism. This is especially interesting and unexpected because, in the light of earlier results, φ rarely deforms to a birational morphism when Y is a rational variety, as is our case. The Fano-K3 case contrasts with canonical morphisms of hyperelliptic curves and with hyperelliptic K3 surfaces of genus g ≥ 3 . Regarding the second topic, we completely answer the question for generalized hyperelliptic polarized Fano and Calabi–Yau varieties. For generalized hyperelliptic varieties of general type we do this in even greater generality, since our result holds for Y toric. Standard methods in deformation theory do not work in the present setting. Thus, to settle these long standing open questions, we bring in new ideas and techniques building on those introduced by the authors concerning deformations of finite morphisms and the existence and smoothings of certain multiple structures. We also prove a new general result on unobstructedness of morphisms that factor through a double cover and apply it to the case of generalized hyperelliptic varieties. [ABSTRACT FROM AUTHOR]

We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations of two new pairs of fake projective plane with $21$ automorphisms, thus finishing the task of finding explicit equations of fake projective planes with this automorphism group. This includes, in particular, the fake projective plane discovered by J. Keum.

We study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123–1158). We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non-Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface S which determine a non-Galois triple cover of S. The examples presented are in any admissible Kodaira dimension, and in particular, we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type. [ABSTRACT FROM AUTHOR]

We study some special systems of generators on finite groups, introduced in previous work by the first author and called diagonal double Kodaira structures, in order to investigate finite non-abelian quotients of the pure braid group on two strands P 2 (Σ b) , where Σ b is a closed Riemann surface of genus b. In particular, we prove that, if a finite group G admits a diagonal double Kodaira structure, then | G | ≥ 32 , and equality holds if and only if G is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two 3-dimensional families of double Kodaira fibrations having signature 16. Such surfaces are different from the ones recently constructed by Lee, Lönne and Rollenske and, as far as we know, they provide the first examples of positive-dimensional families of double Kodaira fibrations with small signature. [ABSTRACT FROM AUTHOR]

We introduce the notion of a simple fibration in $(1,2)$ -surfaces – that is, a hypersurface inside a certain weighted projective space bundle over a curve such that the general fibre is a minimal surface of general type with $p_g=2$ and $K^2=1$ . We prove that almost all Gorenstein simple fibrations over the projective line with at worst canonical singularities are canonical threefolds ‘on the Noether line’ with $K^3=\frac 43 p_g-\frac {10}3$ , and we classify them. Among them, we find all the canonical threefolds on the Noether line that have previously appeared in the literature.

We classify — as far as possible — Gorenstein stable surfaces with K X 2 = 1 and χ (O X) = 2 , describing several strata in the moduli space quite in detail. [ABSTRACT FROM AUTHOR]

Minimal algebraic surfaces of general type X such that K X 2 = 2 χ (O X) - 6 or K X 2 = 2 χ (O X) - 5 are called Horikawa surfaces. In this note we study Z 2 2 -actions on Horikawa surfaces. The main result is that all the connected components of Gieseker's moduli space of canonical models of surfaces of general type with invariants satisfying these relations contain surfaces with Z 2 2 -actions. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to construct a compactification of the moduli space of degree $d \geqslant 5$ surfaces in $\mathbb {P}^{3}_{{{\mathbb {C}}}}$ , i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in $\mathbb {P}^{3}$ and whose boundary points correspond to degenerations of such surfaces. We consider a divisor $D$ on a Fano variety $Z$ as a pair $(Z, D)$ satisfying certain properties. We find a modular compactification of such pairs and, in the case of $Z = {{\mathbb {P}}}^{3}$ and $D$ a surface, use their properties to classify the pairs on the boundary of the moduli space. [ABSTRACT FROM AUTHOR]

In this paper, we investigate algebraic surfaces with Eisenbud–Harris general fibration of genus 4 satisfying a certain equality with respect to the invariants. In particular, we analyze the structure, prove the existence and construct families for these surfaces. Our surfaces are distinguished into several types from some points of view, namely, certain degenerate fibers and their number, and the direct image of the relative dualizing sheaf for the fibration. Furthermore, we obtain some results for the deformation that are related to these distinctions. [ABSTRACT FROM AUTHOR]

In this paper, we study minimal algebraic surfaces of general type with K 2 = 3 p g − 5. Since the case p g ≤ 4 has been studied by Horikawa, Zucconi, and Bauer, we always assume p g ≥ 5 here. We prove that for such a surface, the canonical map is generically finite of degree 1 or 2. If the degree of the canonical map is 2, then the canonical image is a ruled surface, moreover, for any fixed (p g ≥ 14 , K 2 = 3 p g − 5) , there exists such a regular surface. If the degree of the canonical map is 1, then the surface is regular (i.e. q = 0) and its canonical linear system is base point free or has one simple base point. In the case where the canonical map is birational and the canonical linear system is base point free, we study the property of its canonical image and prove that for any fixed (p g ≥ 5 , K 2 = 3 p g − 5) , there exists such a surface. [ABSTRACT FROM AUTHOR]

We consider elliptic surfaces E over a field k equipped with zero section O and another section P of infinite order. If k has characteristic zero, we show there are only finitely many points where O is tangent to a multiple of P. Equivalently, there is a finite list of integers such that if n is not divisible by any of them, then nP is not tangent to O. Such tangencies can be interpreted as unlikely intersections. If k has characteristic zero or p > 3 and E is very general, then we show there are no tangencies between O and nP. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with K ample and K 2 unbounded. [ABSTRACT FROM AUTHOR]

We give two examples of surfaces with canonical map of degree 4 onto a canonical surface.

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 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. [ABSTRACT FROM AUTHOR]