arXiv subjects

Francisco Javier Gallego

Publications and source records attributed to Francisco Javier Gallego.

15 recordsLinked to original sources

arXiv2026

Abelian canonical covers of minimal rational surfaces and $\mathbb F_1$ up to degree $8$

We classify abelian canonical covers of degree at most $8$ of smooth surfaces of minimal degree, and of $\mathbb P^2$ and Hirzebruch surfaces not embedded as surfaces of minimal degree. Let $π: X \longrightarrow W$ be an abelian canonical cover of a smooth surface of minimal degree $W$ with group $G$ and degree $n$. Degrees $n \le 4$ were classified earlier; we complete the picture for $5 \le n \le 8$: - No such covers exist if $n$ is prime and $n > 3$, nor if $G=\mathbb Z_8$. - For $G=\mathbb Z_6$, $\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$ and $\mathbb Z_2 \times \mathbb Z_4$ we give the complete list of covers. Only $W =\mathbb P^2$, $\mathbb F_0$ and $\mathbb F_1$ arise. The list contains several families with unbounded $p_g$, most of them new; $X$ is smooth for general covers with $G=\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$, but singular if $G=\mathbb Z_6$ and, with one exception, if $G=\mathbb Z_2 \times \mathbb Z_4$. When $W$ is $\mathbb P^2$ or a Hirzebruch surface not embedded as a surface of minimal degree, we show that abelian canonical covers exist only for $n = 2$ or for $G =\mathbb Z_2 \times \mathbb Z_2$, and we classify the latter completely: they form infinitely many families, most new, with unbounded $p_g$ and, most of them, with unbounded irregularity. The slopes $K^2/χ$ of all these $X$ accumulate at $6$, at 8, and at $8α/(α+1)$ for every integer $α\ge 2$. These results rest on more general ones, valid for $X$ normal, locally Gorenstein and of arbitrary dimension, and $W$ smooth with $p_g(W) = 0$: we determine which branch divisors of an abelian canonical cover must vanish, and, for canonical covers whose $π_*\mathcal O_X$ splits as a sum of line bundles, we describe the $\mathcal O_W$-module structure of $π_*\mathcal O_X$, computing it completely when $W$ is $\mathbb P^2$ or a Hirzebruch surface.

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arXiv2024

Projective smoothing of varieties with simple normal crossings

In this article, we introduce a new approach to show the existence and smoothing of simple normal crossing varieties in a given projective space. Our approach relates the above to the existence of nowhere reduced schemes called ribbons and their smoothings via deformation theory of morphisms. As a consequence, we prove results on the existence and smoothing of snc subvarieties $V \subset \mathbb{P}^N$, with two irreducible components, each of which are Fano varieties of dimension $n>2$, embedded inside $\mathbb{P}^{N}$ for effective values of $N$, by the complete linear series of a line bundle $H$. The general fibers of the resulting one parameter families are either smooth Fano, Calabi-Yau or varieties of general type, depending on the positivity of the canonical divisor of their intersections. An interesting consequence of projective smoothing is that it automatically gives a smoothing of the semi-log-canonical (slc) pair $(V, Δ)$, where $Δ= cH$, $c < 1$, is a rational multiple of a general hyperplane section of $H$. For threefolds, we are able to give explicit descriptions of the smoothable snc subvarieties due to the classification results of Iskovskikh-Mori-Mukai. In particular, we show the existence of unions V = $Y_1 \bigcup_D Y_2 \subset \mathbb{P}^N$, where $Y_i$'s are smooth anticanonically (resp. bi-anticanonically) embedded Fano threefolds, intersecting along $D$, where $D$ is either a del-Pezzo surface or a $K3$ surface (resp. a smooth surface with ample canonical bundle) and their smoothing in $\mathbb{P}^N$ to smooth Fano or Calabi-Yau threefolds (resp. to threefolds with ample canonical bundle) for various values of $N$ between $10$ and $163$. In cases when the general fiber is a smooth Fano or Calabi-Yau threefold, one can choose $c$ such that $(V, Δ)$ is a Calabi-Yau pair while in all cases $c$ can be chosen so that $(V, Δ)$ is a stable pair.

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arXiv2022

Construction of varieties of low codimension with applications to moduli spaces of varieties of general type

In this article we develop a new way of systematically constructing infinitely many families of smooth subvarieties $X$ of any given dimension $m$, $m \geq 3$, and any given codimension in $\mathbb P^N$, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non--reduced schemes called ropes, embedded in $\mathbb P^N$, and by smoothing them. In the range $3 \leq m < N/2$, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete intersections. We also go beyond a question of Enriques on constructing simple canonical surfaces in projective spaces, and construct simple canonical varieties in all dimensions. The canonical map of infinitely many of these simple canonical varieties is finite birational but not an embedding. Finally, we show the existence of components of moduli spaces of varieties of general type (in all dimensions $m$, $m \geq 3$) that are analogues of the moduli space of curves of genus $g > 2$ with respect to the behavior of the canonical map and its deformations. In many cases, the general elements of these components are canonically embedded and their codimension is in the range of Hartshorne's conjecture.

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arXiv2022

Deformations and moduli of irregular canonical covers with $K^2=4p_g-8$

In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $K_X^2 = 4p_g(X)-8$, for any even integer $p_g\geq 4$. These surfaces also have unbounded irregularity $q$. We carry out our study by investigating the deformations of the canonical morphism $φ:X\to \mathbb{P}^N$, where $φ$ is Galois of degree 4. These canonical covers are classified in by the first two authors into four distinct 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, and consequently, $X$ belongs to a unique irreducible component of the Gieseker moduli space, which we prove is uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality $p_g > 2q-4$, with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. The existence of jumping subloci is a contrast with the moduli of surfaces with $K_X^2 = 2p_g - 4$, studied by Horikawa. There is a similarity and difference to the moduli of curves of genus $g\geq 3$, 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.

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arXiv2021

On higher dimensional extremal varieties of general type

Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type with nef canonical divisor and canonical singularities, whose image $Y$ under the canonical map is of maximal dimension, satisfies $K_X^n \ge 2 (p_g-n)$. We investigate the very interesting extremal situation $K_X^n=2(p_g-n)$, which appears in a number of geometric situations. Since these extremal varieties are natural higher dimensional analogues of Horikawa surfaces, we name them Horikawa varieties. These varieties have been previously dealt with inthe works of Fujita and Kobayashi. We carry out further studies of Horikawa varieties, proving new results on various geometric and topological issues concerning them. In particular, we prove that the geometric genus of those Horikawa varieties whose image under the canonical map is singular is bounded. We give an analogous result for polarized hyperelliptic subcanonical varieties, in particular, for polarized Calabi-Yau and Fano varieties. The pleasing numerology that emerges puts Horikawa's result on surfaces in a broader perspective. We obtain a structure theorem for Horikawa varieties and explore their pluriregularity. We use this to prove optimal results on projective normality of pluricanonical linear systems. We study the fundamental groups of Horikawa varieties, showing that they are simply connected, even if $Y$ is singular. We also prove results on deformations of Horikawa varieties, whose implications on the moduli space make them the higher dimensional analogue of curves of genus $2$.

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arXiv2020

Deformations of hyperelliptic and generalized hyperelliptic polarized varieties

In this article we study the deformations of hyperelliptic polarized varieties $(X,L)$ of dimension $m$ and sectional genus $g$ such that the image $Y$ of the morphism $φ$ induced by $|L|$ is smooth. If $L^m < 2g-2$, it is known that, by adjunction and the Clifford's theorem, any deformation of $(X,L)$ is hyperelliptic. Thus, we focus on when $L^m=2g-2$ or $L^m=2g$. We prove that, if $(X,L)$ is Fano-K3, then, except when $Y$ is a hyperquadric, all deformations of $(X,L)$ are again hyperelliptic (if $Y$ is a hyperquadric, the general deformation of $φ$ is an embedding). This contrasts with the situation of hyperelliptic canonical curves and hyperelliptic K3 surfaces. If $L^m=2g$, then we prove that, in most cases, a general deformation of $φ$ is a finite morphism of degree $1$. This provides interesting examples of degree $2$ morphisms that can be deformed to morphisms of degree $1$. We extend our results to so-called generalized hyperelliptic polarized Fano, Calabi-Yau and general type varieties. The solutions to these questions are closely intertwined with the existence or non existence of double structures on the algebraic varieties $Y$. We address this matter as well.

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arXiv2015

Deformations of canonical double covers

In this paper, we show that if $X$ is a smooth variety of general type of dimension $m \geq 2$, for which its canonical map induces a double cover onto $Y$, where $Y$ is a projective bundle over $\mathbf P^1$, or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series, then the general deformation of the canonical morphism of $X$ again is canonical and again induces a double cover. The second part of the article deals with the existence or non existence of canonical double structures on rational varieties. The negative result in this article has consequences for the moduli of varieties of general type of arbitrary dimension. The results here show that there is an entire component, that is hyperelliptic in infinitely many moduli spaces of higher dimensional varieties of general type. This is in sharp contrast with the case of curves or surfaces of lower Kodaira dimensions.

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arXiv2015

Deformations of canonical triple covers

In this paper, we show that if $X$ is a smooth variety of general type of dimension $m \geq 3$ for which the canonical map induces a triple cover onto $Y$, where $Y$ is a projective bundle over $\mathbf P^1$ or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series, then the general deformation of the canonical morphism of $X$ is again canonical and induces a triple cover. The extremal case when $Y$ is embedded as a variety of minimal degree is of interest, due to its appearance in numerous situations. This is especially interesting as well, since it has no lower dimensional analogues.

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arXiv2012

Smoothable locally Cohen--Macaulay and non Cohen--Macaulay multiple structures on curves

In this article we show that a wide range of multiple structures on curves arise whenever a family of embeddings degenerates to a morphism $φ$ of degree $n$. One could expect to see, when an embedding degenerates to such a morphism, the appearance of a locally Cohen-Macaulay multiple structure of certain kind (a so-called rope of multiplicity $n$). We show that this expectation is naive and that both locally Cohen-Macaulay and non Cohen-Macaulay multiple structures might occur in this situation. In seeing this we find out that many multiple structures can be smoothed. When we specialize to the case of double structures we are able to say much more. In particular, we find numerical conditions, in terms of the degree and the arithmetic genus, for the existence of many smoothable double structures. Also, we show that the existence of these double structures is determined, although not uniquely, by the elements of certain space of vector bundle homomorphisms, which are related to the first order infinitesimal deformations of $φ$. In many instances, we show that, in order to determine a double structure uniquely, looking merely at a first order deformation of $φ$ is not enough; one needs to choose also a formal deformation.

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arXiv2010

An infinitesimal condition to deform a finite morphism to an embedding

In this article we give a sufficient condition for a morphism $φ$ from a smooth variety $X$ to projective space, finite onto a smooth image, to be deformed to an embedding. This result puts some theorems on deformation of morphisms of curves and surfaces such as $K3$ and general type, obtained by ad hoc methods, in a new, more conceptual light. One of the main interests of our result is to apply it to the construction of smooth varieties in projective space with given invariants. We illustrate this by using our result to construct canonically embedded surfaces with $c_1^2=3p_g-7$ and derive some interesting properties of their moduli spaces. Another interesting application of our result is the smoothing of ropes. We obtain a sufficient condition for a rope embedded in projective space to be smoothable. As a consequence, we prove that canonically embedded carpets satisfying certain conditions can be smoothed. We also give simple, unified proofs of known theorems on the smoothing of $1$--dimensional ropes and $K3$ carpets. Our condition for deforming $φ$ to an embedding can be stated very transparently in terms of the cohomology class of a suitable first order infinitesimal deformation of $φ$. It holds in a very general setting (any $X$ of arbitrary dimension and any $φ$ unobstructed with an algebraic formally semiuniversal deformation). The simplicity of the result can be seen for instance when we specialize it to the case of curves.

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arXiv2010

On the deformations of canonical double covers of minimal rational surfaces

The purpose of this article is to study the deformations of smooth surfaces $X$ of general type whose canonical map is a finite, degree 2 morphism onto a minimal rational surface or onto $\mathbf F_1$, embedded in projective space by a very ample complete linear series. Among other things, we prove that any deformation of the canonical morphism of such surfaces $X$ is again a morphism of degree 2. A priori, this is not at all obvious, for the invariants $(p_g(X),c_1^2(X))$ of most of these surfaces lie on or above the Castelnuovo line; thus, in principle, a deformation of such $X$ could have a birational canonical map. We also map the region of the geography of surfaces of general type corresponding to the invariants of the surfaces $X$ and we compute the dimension of the irreducible moduli component containing $[X]$. In certain cases we exhibit some interesting moduli components parametrizing surfaces $S$ whose canonical map has different behavior but whose invariants are the same as the invariants of $X$. One of the interests of the article is that we prove the results about moduli spaces employing crucially techniques on deformation of morphisms. The key point or our arguments is the use of a criterion that requires only infinitesimal, cohomological information of the canonical morphism of $X$. As a by-product, we also prove the non-existence of "canonically" embedded multiple structures on minimal rational surfaces and on $\mathbf F_1$.

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arXiv2006

K3 double structures on Enriques surfaces and their smoothings

Let $Y$ be a smooth Enriques surface. A $K3$ carpet on $Y$ is a locally Cohen-Macaulay double structure on $Y$ with the same invariants as a smooth $K3$ surface (i.e., regular and with trivial canonical sheaf). The surface $Y$ possesses an étale $K3$ double cover $X \oversetπ \longrightarrow Y$. We prove that $π$ can be deformed to a family $\SX \longrightarrow \mathbf P^N_{T^*}$ of projective embeddings of $K3$ surfaces and that any projective $K3$ carpet on $Y$ arises from such a family as the flat limit of smooth, embedded $K3$ surfaces.

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arXiv2003

Classification of quadruple Galois canonical covers II

In this article we classify quadruple Galois canonical covers $ϕ$ of singular surfaces of minimal degree. This complements the work done in math.AG/0302045, so the main output of both papers is the complete classification of quadruple Galois canonical covers of surfaces of minimal degree, both singular and smooth. Our results show that the covers $X$ studied in this article are all regular surfaces and form a bounded family in terms of geometric genus $p_g$. In fact, the geometric genus of $X$ is bounded by 4. Together with the results of Horikawa and Konno for double and triple covers, a striking numerology emerges that motivates some general questions on the existence of higher degree canonical covers. In this article, we also answer some of these questions. The arguments to prove our results include a delicate analysis of the discrepancies of divisors in connection with the ramification and inertia groups of $ϕ$.

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arXiv2002

Triple canonical covers of varieties of minimal degree

In this article we study pluriregular varieties X of general type with base-point-free canonical bundle whose canonical morphism has degree 3 and maps X onto a variety of minimal degree Y. We carry out our study from two different perspectives. First we study in Section 2 the canonical ring of X describing completely the degrees of its minimal generators. We apply this to the study of the projective normality of the images of the pluricanonical morphisms of X. Our study of the canonical ring of $X$ also shows that, if the dimension of X is greater than or equal to 3, there does not exist a converse to a theorem of M. Green that bounds the degree of the generators of the canonical ring of X. This is in sharp contrast with the situation in dimension 2 where such converse exists, as proved by the authors in a previous work. Second, we study in Section 3, the structure of the canonical morphism of X. We use this to show among other things the non-existence of some a priori plausible examples of triple canonical covers of varieties of minimal degree. We also characterize the targets of flat canonical covers of varieties of minimal degree. Some of the results of Section 3 are more general and apply to varieties X which are not necessarily regular, and to targets Y that are scrolls which are not of minimal degree.

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arXiv2000

Some results on rational surfaces and Fano varieties

The goal of this article is to study the equations and syzygies of embeddings of rational surfaces and certain Fano varieties. Given a rational surface X and an ample and base-point-free line bundle L on X, we give an optimal numerical criterion for L to satisfy property Np. This criterion turns out to be a characterization of property Np if X is anticanonical. We also prove syzygy results for adjunction bundles and a Reider type theorem for higher syzygies. For certain Fano varieties we also prove results on very ampleness and higher syzygies.

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