An Effective Criterion for Covering Maps Between Real Varieties
Peking University（北京大学）
Rizeng Chen
备注：
Revised. The proof of Theorem 3.3 is greatly simplified. Expand the section on algorithms. 34 pages, 10 figures. This supersedes arXiv:2502.05834 as the results are largely improved and expanded
We prove that a quasi-finite flat morphism with locally constant geometric-fiber cardinality between varieties over a real closed field induces a covering map on the rational points. The algebraic core of the criterion is a characteristic-zero result showing that every such morphism becomes finite étale after reduction. Since reduction does not change rational points, the induced map is a covering in the Euclidean topology. We extend the covering conclusion to Boolean combinations of closed subschemes and their complements, and construct a finite stratification separating finite covering families from families with positive-dimensional fibers. Finally, we compute the canonical non-finite, non-finite-flat, and non-finite-étale loci, using Gröbner bases. This yields effective tests for the hypotheses of the covering criterion. We conclude with applications illustrating these constructions.
Let $f:X\to Y$ be a finite ramified Galois covering of algebraic varieties defined over the complex numbers. In this paper, we prove some structure theorems for such coverings in the case that the non-abelian Galois group of the cover is metabelian (in particular metacyclic). Our results extend the previous ones, obtained by several authors, in the case of abelian and dihedral Galois coverings. In analogy with the abelian (and generalizing the dihedral) case, we find "building data" of metabelian covers from which the cover can be reconstructed. In addition to analyzing several examples, we co…
Under the condition that the Prym map is injective in characteristic $p$, we prove that the special subvarieties in the moduli space of abelian varieties of dimension $l$ and polarization type $D$, $A_{l,D}$, arising from families of abelian covers of $¶^1$ are of a very restrictive nature. In other words, if the family is one-dimensional or if it contains an eigenspace of certain type for the group action on the cohomology of fibers, then the Shimura varieties arising from such families can only be constructed by the group action of the family.
It is well known that the Prym variety of an étale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny is an isomorphism. We compute the degree of this isogeny in the remaining cases and show that only in the case of coverings of degree 4 it is an isomorphism.
We suggest a combinatorial criterion for the smoothness of an arbitrary spherical variety using the classification of multiplicity-free spaces, generalizing an earlier result of Camus for spherical varieties of type $A$.
Given a family of Galois coverings of the projective line we give a simple sufficient condition ensuring that the closure of the image of the family via the period mapping is a special (or Shimura) subvariety in A_g. By a computer program we get the list of all families in genus up to 8 satisfying our condition. There is no family in genus 8, all of them are in genus at most 7. These examples are related to a conjecture of Oort. Among them we get the cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot, Rohde, Moonen and others) and the abelian non-cyclic examples foun…
We define the covering gonality and separable covering gonality of varieties over arbitrary fields, generalizing the definition given by Bastianelli-de Poi-Ein-Lazarsfeld-Ullery for complex varieties. We show that over an arbitrary field a smooth multidegree $(d_1,\ldots,d_k)$ complete intersection in $\mathbb{P}^N$ has separable covering gonality at least $d-N+1$, where $d=d_1+\cdots+d_k$. We also show that the very general such variety has covering gonality at least $\frac{d-N+2}{2}$.
G. Pareschi and M. Popa give criterions for global generations and surjectivity of multiplication maps of global sections of coherent sheaves on abelian varieties in the theory of M-regularity. In this paper, we generalize some of their criterions via the M-regularity of $\mathbb{Q}$-twisted sheaves. As an application, we show that the M-regularity of a suitable $\mathbb{Q}$-twisted sheaf implies property $(N_p)$ and jet-ampleness for ample line bundles on abelian varieties.
We study the Prym varieties arising from étale cyclic coverings of degree 7 over a curve of genus 2. These Prym varieties are products of Jacobians JY x JY of genus 3 curves Y with polarization type D=(1,1,1,1,1,7). We describe the fibers of the Prym map between the moduli space of such coverings and the moduli space of abelian sixfolds with polarization type D, admitting an automorphism of order 7.
We study a conjecture, due to Voisin, on 0-cycles on varieties with $p_g=1$. Using Kimura's finite dimensional motives and recent results of Vial's on the refined (Chow-)Künneth decomposition, we provide a general criterion for Calabi-Yau manifolds of dimension at most $5$ to verify Voisin's conjecture. We then check, using in most cases some cohomological computations on the mirror partners, that the criterion can be successfully applied to various examples in each dimension up to $5$.
Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\mbox{Prym}(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\mbox{Prym}(\widetilde{C}_n/C_n)$ as a Prym va…
We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building on the work of Pirola, we show that very general abelian varieties of dimension $g$ have covering gonality $k\geq f(g)$ where $f(g)$ grows like ${\rm log}\,g$. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties, eg. if $g\geq 2k-1$, for any divisor $D\in {\rm Pic}^…
We propose a simple criterion to know if an abelian variety $A$ defined over a finite field $\mathbb{F}_q$ is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring End$_{\mathbb{F}_q}(A)$. We also provide a criterion to know if an isogeny class is cyclic, i.e., all its varieties are cyclic; this criterion is based on the characteristic polynomial of the isogeny class. We find some asymptotic lower bounds on the fraction of cyclic $\mathbb{F}_q$-isogeny classes among certain families of them, when $q$ tends to infinity. Some of these bounds req…
We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the c…
We prove a stronger version of a criterion of rationality given by Ionescu and Russo. We use this stronger version to define an invariant for rational varieties (we call it rationality degree), and we classify rational varieties for small values of the invariant.
We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, we show that for simple abelian varieties a better bound is possible. As an application of these results we show that if $E$ is an elliptic curve over a finite field then for any $n\in \mathbb{N}$ there exist smooth curves of bounde…
