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In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $\kappa(S)=1$), in the initial case $ \chi(\mathcal{O}_S) =0$. In particular, in the case where $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2$. We also show with easy examples that the groups $\mathbb{Z}/2, \mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components., Comment: 49 pages, to appear in a volume of the Taiwanese Journal of Mathematics dedicated to Yurii (Gennadievich) Prokhorov on the occasion of his 60th birthday

In this paper we characterize the compact orbifolds, quotients $ X = \mathcal{D} /\Gamma$ of a bounded symmetric domain $ \mathcal{D}$ of tube type by the action of a discontinuous group $\Gamma$, as those projective orbifolds with ample canonical divisor possessing a slope zero tensor of `orbifold type'., Comment: 13 pages, dedicated to the memory of Lucian Badescu, to appear in the Revue Roumaine Math. Pures Appl. arXiv admin note: text overlap with arXiv:2403.06044. Missing definition of orbifold canonical divisor added

In this paper we characterize the quotients $ X = T/G$ of a complex torus $T$ by the action of a finite group $G$ as the K\"ahler orbifold classifying spaces of the even Euclidean cristallographic groups $\Gamma$, and we prove other similar and stronger characterizations., Comment: 11 pages, submitted to Rendiconti Semin. Mat. Univ.Politecn. Torino, volume dedicated to the memory of Gianfranco Casnati

We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of rational maps, via specializations. Among the applications is a new proof of a result obtained jointly with Luca Cesarano: that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $\Phi_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal. The proof is also based on a careful study of the Theta divisors of the Jacobians of Hyperelliptic curves, and some related geometrical constructions. We investigate these here also in view of their beauty and of their independent interest, as they lead to a description of the rings of Hyperelliptic theta functions., Comment: 20 pages. The revision amends two incorrect statements: the author is extremely thankful to the referee for pointing out two obscure arguments, which were indeed incorrect, and needed a reparation

We consider the geometric map $ \mathfrak C$, called Cayleyan, associating to a plane cubic $E$ the adjoint of its dual curve. We show that $ \mathfrak C$ and the classical Hessian map $ \mathfrak H$ generate a free semigroup. We begin the investigation of the geometry and dynamics of these maps, and of the geometrically special elliptic curves: these are the elliptic curves isomorphic to cubics in the Hesse pencil which are fixed by some endomorphism belonging to the semigroup $\mathcal W(\frak H, \frak C)$ generated by $ \frak H, \frak C$. We point out then how the dynamic behaviours of $ \mathfrak H$ and $ \mathfrak C$ differ drastically. Firstly, concerning the number of real periodic points: for $ \mathfrak H$ these are infinitely many, for $ \mathfrak C$ they are just $4$. Secondly, the Julia set of $ \mathfrak H$ is the whole projective line, unlike what happens for all elements of $\mathcal W (\frak H, \frak C)$ which are not iterates of $ \mathfrak H$., Comment: 26 pages

Motivated by a general question addressed by Mario Baldassarri in 1956, we discuss characterizations of the Pseudo-Abelian Varieties introduced by Roth, and we introduce a first new notion, of Manifolds Isogenous to a k-Torus Product: the latter have the last k Chern classes trivial in rational cohomology and vanishing Chern numbers. We show that in dimension 2 the latter class is the correct substitute for some incorrect assertions by Enriques, Dantoni, Roth and Baldassarri: these are the surfaces with $K_X$ nef and $c_2(X)=0 \in H^4(X, \mathbb{Z})$. We observe in the last section, using a construction by Chad Schoen, that such a simple similar picture does not hold in higher dimension. We discuss then, as a class of solutions to Baldassarri's question, manifolds isogenous to projective (respectively: K\"ahler) manifolds whose tangent bundle or whose cotangent bundle has a trivial subbundle of positive rank. We see that the class of `partially framed' projective manifolds (that is, whose tangent bundle has a trivial subbundle) consists, in the case where $K_X$ is nef, of the Pseudo-Abelian varieties of Roth; while the class of `partially co-framed' projective manifolds is not yet fully understood in spite of the new results that we are able to show here: and we formulate some open questions and conjectures. In the course of the paper we address also the case of more general compact complex Manifolds, introducing the new notions of suspensions over parallelizable Manifolds, of twisted hyperelliptic Manifolds, and we describe the known results under the K\"ahler assumption., Comment: 31 pages, the revised II Part contains new results, in particular a much stronger Theorem 1.5; it treats also the non K\"ahler case. Misprint in the introduction removed. arXiv admin note: substantial text overlap with arXiv:2206.02646

In this short note we prove two theorems, the first one is a sharpening of a result of Lange and Sernesi: the discriminant curve W of a general Abelian surface $A$ endowed with an irreducible polarization $D$ of type $(1,3)$ is an irreducible curve of degree $18$ whose singularities are exactly $36$ nodes and $72$ cusps. Moreover, we analyze the degeneration of the discriminant curve $W$ and its singularities as $A$ tends to the product of two equal elliptic curves. The second theorem, using the first one in order to prove a transversality assertion, shows that the general element of a family of surfaces constructed by Alessandro and Catanese is a smooth surface, thereby proving the existence of a new family of minimal surfaces of general type with $p_g=q=2, K^2=6$ and Albanese map of degree $3$., Comment: 13 pages, dedicated to the memory of Alberto Collino

We give first an easy construction of surfaces with $p_g=q=2, K^2=5$ and Albanese map of degree $3$, describing an irreducible connected component of the moduli space of surfaces of general type, which we show to be the only one of the Main Stream with these invariants and satisfying a mild condition. We call it the family of CHPP surfaces, since it contains the family constructed by Chen and Hacon, and coincides with the one considered by Penegini and Polizzi. We also give an easy construction of an irreducible connected component of the moduli space of surfaces of general type with $p_g=q=2, K^2=6$ and Albanese map of degree $4$, which we call the family of PP4 surfaces since it contains the family constructed by Penegini and Polizzi. Finally, we answer a question posed by Chen and Hacon, via three families of surfaces with $p_g=q$ whose Tschirnhaus module has a kernel realization with quotient a nontrivial homogeneous bundle. Two families have $p_g=q=3$, the third is a new family of surfaces with $p_g=q=2, K^2=6$ and Albanese map of degree $3$ (the existence of this family is based on arXiv:2212.14877, joint work of the second author with Edoardo Sernesi) ., Comment: 44 pages, to appear in a volume dedicated to the memory of Alberto Collino. Small improvements added

We attach two binary codes to a projective nodal surface (the strict code K and, for even degree d, the extended code K' ) to investigate the `Nodal Severi varieties F(d, n) of nodal surfaces in P^3 of degree d and with n nodes, and their incidence hierarchy, relating partial smoothings to code shortenings. Our first main result solves a question which dates back over 100 years: the irreducible components of F(4, n) are in bijection with the isomorphism classes of their extended codes K', and these are exactly all the 34 possible shortenings of the extended Kummer code K' , and a component is in the closure of another if and only if the code of the latter is a shortening of the code of the former. We extend this result classifying the irreducible components of all nodal K3 surfaces in the same way, and we fully classify their extended codes. In this classification there are some sporadic cases, obtain through projection from a node. For surfaces of degree d=5 in P^3 we determine (with one possible exception) all the possible codes K, and for several cases of K, we show the irreducibility of the corresponding open set of F(5, n), for instance we show the irreducibility of the family of Togliatti quintic surfaces. In the fourth part we show that a `Togliatti-like' description holds for surfaces of degree 6 with the maximum number of nodes= 65: they are discriminants of cubic hypersurfaces in P^6 with 31 (respectively 32) nodes, and we have an irreducible 18-dimensional family of them. For degree d=6, our main result is based on some novel auxiliary results: 1) the study of the half-even sets of nodes on sextic surfaces, 2) the investigation of discriminants of cubic hypersurfaces X, 3) the computer assisted proof that, for n = 65, both codes K, K' are uniquely determined, 4) the description of these codes, relating the geometry of the Barth sextic with the Doro-Hall graph., Comment: 210 pages. v2 minor changes to text, added ancillary files containing supporting computer code. The new version contains some correction, completes some linear algebra proofs of code classification, adds a table showing the Kummer genealogy tree of nodal quartic surfaces. Adds also new references

We describe examples showing the sharpness of Fujita's conjecture on adjoint bundles also in the general type case, and use these examples to formulate related bold conjectures on pluricanonical maps of varieties of general type., Comment: 6 pages, to appear in the Journal `Complex Manifolds'

We show that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 12, if the minimal resolution of $X$ is not a supersingular K3 surface. We also provide a family of explicit examples, valid in any characteristic., Comment: 21 pages

We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only projective manifolds with trivial Chern classes. By Yau' s celebrated result, compact K\"ahler manifolds with trivial Chern classes must be flat, that is, they belong to the class of Hyperelliptic Manifolds (quotients $T/G$ of a complex torus $T$ by the free action of a finite group $G$). We exhibit simple examples of projective Hyperelliptic Manifolds which are not Abelian varieties and whose Chern classes are zero not only in integral cohomology, but also in the Chow ring. We prove moreover that the Bagnera-de Franchis manifolds (quotients $T/G$ as above but where the group $ G$ is cyclic) have topologically trivial tangent bundle. Our results naturally lead to the question of classifying all compact K\"ahler manifolds with topologically trivial tangent bundle, and all the counterexamples to Severi's question., Comment: 17 pages, the previous article which is here replaced has been split into 2 parts, separating the question by Severi and the question by Baldassarri in order to achieve more clarity of presentation. Small alterations are also included, dedicated to the memory of Mario Baldassarri. Part II has also been uploaded on arXiv

We show, in this second part, that the maximal number of singular points of a quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 14, and that, if we have 14 singularities, these are nodes and moreover the minimal resolution of $X$ is a supersingular K3 surface. We produce an irreducible component, of dimension 24, of the variety of quartics with 14 nodes. We also exhibit easy examples of quartics with 7 $A_3$-singularities., Comment: v2: 35 pages, final version with improved exposition and streamlined arguments; dedicated to Herb Clemens on occasion of his 82nd birthday

We describe the birational and the biregular theory of cyclic and Abelian coverings between real varieties., Comment: 25 pages, dedicated to Slava (Viatcheslav) Kharlamov on the occasion of his 71-st birthday, final version to appear on Mathematische Annalen

We show that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic $2$ is at most $20$, and that if equality is attained, then the minimal resolution of $X$ is a supersingular K3 surface and the singular points are $20$ nodes. We produce examples with 14 nodes. In a sequel to this paper (in two parts, the second in collaboration with Matthias Sch\"utt) we show that the optimal bound is indeed 14, and that if equality is attained, then the minimal resolution of $X$ is a supersingular K3 surface and the singular points are $14$ nodes. We also obtain some smaller upper bounds under several geometric assumptions holding at one of the singular points $P$ (structure of tangent cone, separability/inseparability of the projection with centre $P$)., Comment: 31 pages. Essentially superseded by the following paper in two parts, entitled `Singularities of normal quartic surfaces I, II (char=2)'; some folklore or side results are used in the next paper

In this paper we first show that each Kummer quartic surface (a quartic surface $X$ with 16 singular points) is, in canonical coordinates, equal to its dual surface, and that the Gauss map induces a fixpoint free involution $\gamma$ on the minimal resolution $S$ of $X$. Then we study the corresponding Enriques surfaces $S/ \gamma$. We also describe in detail the remarkable properties of the most symmetric Kummer quartic, which we call the Cefal\'u quartic. We also investigate the Kummer quartic surfaces whose associated Abelian surface is isogenous to a product of elliptic curves through an isogeny with kernel $(\mathbb{Z}/2)^2$, and show the existence of polarized nodal K3 surfaces $X$ of any degree $d=2k$ with the maximal number of nodes, such that $X$ and its nodes are defined over $\mathbb{R}$. We take then as parameter space for Kummer quartics an open set in $\mathbb{P}^3$, parametrizing nondegenerate $(16_6, 16_6)$-configurations, and compare with other parameter spaces. We also extend to positive characteristic some results which were previously known over $\mathbb{C}$. We end with a section devoted to remarks on normal cubic surfaces, and providing some other examples of strictly selfdual hypersurfaces., Comment: 38 pages, dedicated to Ciro Ciliberto on the occasion of his 70th birthday, to appear in a dedicated volume edited by Springer. In the revised version some questions are answered, which were left open in the first version. We also give a proof in all char $\neq 2$ that each Kummer quartic has infinite automorphism group

In this paper, we investigate automorphisms of compact K\"ahler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of $C^\infty$-isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected components which can be arbitrarily large., Comment: 32 pages; fixed a small gap in Theorem 2.2; final version; to appear in Issue of the Journal "Rendiconti Lincei" dedicated to the memory of Prof. E. Vesentini

The main theorem of this paper is that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $\Phi_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal (i.e., its Pfaffian $d$ is not equal to $1$). We also show that, setting $g = dim (A)$, and letting $d$ be the Pfaffian of the polarization given by $X$, then if $X$ is smooth and $$\Phi_X : X \rightarrow \mathbb{P}^{N:=g+d-2}$$ is an embedding, then necessarily we have the inequality $ d \geq g + 1$, equivalent to $N : = g+d-2 \geq 2 \ dim(X) + 1.$ We also formulate the following interesting conjecture, motivated by work of the second author: if $ d \geq g + 1,$ then, for a general pair $(A,X)$, $\Phi_X$ is an embedding., Comment: 11 pages, dedicated to Olivier Debarre on the occasion of his 60th birthday. Final verion, to appear in a special volume of ERA dedicatee to birational geometry

We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group $Sp (2g, \mathbb{Z})$., Comment: 26 pages, new examples added

We give a characterizaton of smooth ample Hypersurfaces in Abelian Varieties and also describe an irreducible connected component of their moduli space: it consists of the Hypersurfaces of a given polarization type, plus the iterated univariate coverings of normal type (of the same polarization type). The above manifolds yield also a connected component of the open set of Teichm\"uller space consisting of K\"ahler complex structures., Comment: 17 pages, final version to appear in Advances in Math

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities. A concrete application is for surfaces with geometric genus $p_g=5$: the canonical model is embedded in $\mathbb{P}^4$ if and only if we have a complete intersection of type $(2,4)$ or $(3,3)$., Comment: 22 pages

We describe the possible linear actions of a cyclic group $G = \mathbb{Z} /n$ on a complex torus, using the cyclotomic exact sequence for the group algebra $\mathbb{Z} [G]$. The main application is devoted to a structure theorem for Bagnera-De Franchis Manifolds, but we also give an application to hypergeometric integrals., Comment: 23 pages

The Del Pezzo surface $Y$ of degree 5 is the blow up of the plane in 4 general points, embedded in $\mathbb{P}^5$ by the system of cubics passing through these points. It is the simplest example of the Buchsbaum-Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $\mathfrak S_5$. We give canonical explicit $\mathfrak S_5$-invariant Pfaffian equations through a $6 \times 6$ antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $\mathfrak S_5$. Finally, we give $\mathfrak S_5$-invariant equations for the embedding of $Y$ inside $(\mathbb{P}^1)^5$, and show that they have the same Hilbert resolution as for the Del Pezzo of degree $4$., Comment: 25 pages, more explicit formulae and pictures added. Final version to appear in the Rendiconti Circ. Mat. Palermo

We complete the classification of hyperelliptic threefolds, describing in an elementary way the hyperelliptic threefolds with group $D_4$. These are algebraic and form an irreducible 2-dimensional family. Our paper is fully self-contained., Comment: 6 pages

We give a simple construction for the hyperelliptic threefolds with group $D_4$, thus completing the classification of hyperelliptic threefolds., Comment: 4 pages

We prove the equisingular rigidity of the singular Hirzebruch-Kummer coverings $X(n, \mathcal{L})$ of the projective plane branched on line configurations $\mathcal{L}$, satisfying some technical condition. In the case, $\mathcal{L}$ = the complete quadrangle, we give explicit equations of the Hirzebruch-Kummer covering $S_n$ (=the minimal desingularisation of $X(n, \mathcal{L})$) in a product of four Fermat curves of degree n. Since $S_n$ is the $(\mathbb{Z}/n)^5$ covering of the Del Pezzo surface $Y_5$ of degree 5 branched on the 10 lines, these equations are derived from explicit equations of the image of $Y_5$ in $(\mathbb{P}^1)^4$. Version2: We added a new section, describing more generally determinantal equations for all Del Pezzo surfaces of degree $9-k \leq 6$ as subvarieties of the k-fold product of the projective line., Comment: 21 pages

We show, for several fake projective planes with nontrivial automorphism group, that the bicanonical map is an embedding., Comment: A new section (Section 6) is added to point out that, due to many wrong proofs, the main result of the paper by S.-K. Yeung [Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients, Forum Math. 2017;30(2): 419-432] is not proven at all

In this note we give a detailed proof of a result (sketched by Torsten Ekedahl in a discussion with the first author several years ago), describing complex tori admitting a rigid group action and showing explicitly their projectivity and their structure in terms of CM-fields. In the appendix, joint with Benoit Claudon, we show, using a method of Green-Voisin, that all group actions on complex tori deform to projective ones., Comment: 13 pages, new version where the incorrect old propositions 8 and 20 are removed. To appear in Comm. Contemp. Math

We generalize a result of Fujita, on the decomposition of Hodge bundles over curves, to the case of a higher dimensional base., Comment: 10 pages

Motivated by the theory of Inoue-type varieties, we give a structure theorem for projective manifolds $W_0$ with the property of admitting a 1-parameter deformation where $W_t$ is a hypersurface in a projective smooth manifold $Z_t$. Their structure is the one of special iterated univariate coverings which we call of normal type, which essentially means that the line bundles where the univariate coverings live are tensor powers of the normal bundle to the image $X$ of $W_0$. We give applications to the case where $Z_t$ is projective space, respectively an Abelian variety., Comment: 14 pages, final version to appear on Journal Mathematiques Pures Appliquees (Journal de Liouville)

In this paper we achieve a description of the connected components of Teichm\"uller space corresponding to Generalized Hyperelliptic Manifolds $X$. These are the quotients $ X = T/G$ of a complex torus $T$ by the free action of a finite group $G$, and they are also the K\"ahler classifying spaces for a certain class of Euclidean cristallographic groups $\Gamma$, the ones which are torsion free and even., Comment: Final version, appeared in the Proceedings of the Conference `Complex and Symplectic Geometry' (Cortona, June 2016), published by Springer Lecture Notes in Mathematics series of INdAM

We give new contributions to the existence problem of canonical surfaces of high degree. We construct several families (indeed, connected components of the moduli space) of surfaces $S$ of general type with $p_g=5,6$ whose canonical map has image $\Sigma$ of very high degree, $d=48$ for $p_g=5$, $d=56$ for $p_g=6$. And a connected component of the moduli space consisting of surfaces $S$ with $K^2_S = 40, p_g=4, q=0$ whose canonical map has always degree $\geq 2$, and, for the general surface, of degree $2$ onto a canonical surface $Y$ with $K^2_Y = 12, p_g=4, q=0$. The surfaces we consider are SIP 's, i.e. surfaces $S$ isogenous to a product of curves $(C_1 \times C_2 )/ G$; in our examples the group $G$ is elementary abelian, $G = (\mathbb{Z}/m)^k$. We also establish some basic results concerning the canonical maps of any surface isogenous to a product, basing on elementary representation theory., Comment: 29 pages, submitted to a volume dedicated to Lawrence Ein on the occasion of his 60th birthday

The main goal of this paper is to show that Castelnuovo- Enriques' $P_{12}$-theorem also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ($char(k) = p > 0$). The $P_{12}$-theorem is a precise version of the rough classification of algebraic surfaces, in particular the conditions $P_{12} = 0$, $P_{12} = 1$,$P_{12} \geq 2$ are respectively equivalent to : Kodaira dimension $-\infty, 0 , \geq 1$. The result relies on a main theorem describing the growth of the plurigenera for properly-elliptic or properly quasi-elliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e. the families of surfaces which show that the results of the main theorem are sharp., Comment: 19 pages; added some remark, for instance that for complex non algebraic surfaces one needs the plurigenus $P_{42}$