On surfaces of high degree with respect to the sectional genus
We study and classify linearly normal surfaces in $\mathbf{P}^n$, of degree $d$ and sectional genus $g$, such that $d\geq 2g-1$.
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arXiv subjects
Margarida Mendes Lopes
Publications and source records attributed to Margarida Mendes Lopes.
The parity of theta characteristics is preserved by infinitesimal deformations
In this note, given a family of relative dimension one over a smooth curve, we determine the parity of the restriction of a relative theta characteristic to an arbitrary multiple of a fiber in terms of the parity of the restriction to a general fibre. This result can be regarded as a variant of the well-known theorem on the invariance of the parity of theta characteristics in families. As a corollary, we obtain that the torsion subsheaf of the first higher direct image sheaf of a relative theta characteristic splits as a direct sum of two isomorphic sheaves.
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Surfaces with canonical map of odd degree
Let $S$ be a smooth complex minimal surface of general type with $p_g:=h^0(K_S)\ge 4$ whose canonical map is generically finite of odd degree $d>1$ onto a surface $Σ$. We assume that the general canonical curve of $S$ is smooth and that $Σ$ is ruled by lines, and we prove: - $p_g\le d+2$ - $Σ$ is a cone over the rational normal curve of degree $p_g-2$ in ${\mathbb P}^{p_g-1}$ - $p_g=d+2$ can occur only for $d=3,9,11$. As a byproduct, we refine previous results by Beauville and Xiao by proving that if one drops the assumption that $Σ$ is ruled by lines then $d\le 5$ if $p_g\ge 112$. The case $d=3$ being completely classified by the first two named authors, we focus on $d=5$, showing that $p_g\le 5$ and that for $p_g=5$ the surface $S$ has a pencil $|C|$ with $C^2=1$ and $K_SC=5$. These results suggest that the answer to the question whether the surfaces with canonical map of odd degree $d>1$ have bounded invariants could be positive, in sharp contrast with the case of even degree.
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Towards the classification of semistable fibrations having exactly five singular fibers
Let X be a non singular projective surface. Given a semistable non isotrivial fibration f over a smooth rational curve with general fiber non hyperelliptic of genus g bigger than 3, we show that if the number s of singular fibers is 5, then the genus is less or equal to 11, thus improving the previously known bound (g lesser or equal to 17). Furthermore we show that for each possible genus the general fiber has gonality at most 5 and we describe the corresponding fibrations as the resolution of concrete pencils of curves on minimal rational surfaces.
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A footnote to a theorem of Kawamata
Kawamata has shown that the quasi-Albanese map of a quasi-projective variety with log-irregularity equal to the dimension and log-Kodaira dimension 0 is birational. In this note we show that under these hypotheses the quasi-Albanese map is proper in codimension 1 as conjectured by Iitaka.
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Effective characterization of quasi-abelian surfaces
Let V be a smooth quasi-projective complex surface such that the three first logarithmic plurigenera are equal to 1 and the logarithmic irregularity is equal to 2. We prove that the quasi-Albanese morphism of V is birational and there exists a finite set S such that the quasi-Albanese map is proper over the complement of S in the quasi-Albanese variety A(V) of V. This is a sharp effective version of a classical result of Iitaka.
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Numerical properties of exceptional divisors of birational morphisms of smooth surfaces
We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.
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On the degree of the canonical map of a surface of general type
Let X be a minimal complex surface of general type such that its image via the canonical map is a surface; we denote by d the degree of the canonical map. In this expository work, first of all we recall the known possibilities for the canonical image and for d when the canonical map is not birational, which are quite a few, and then we consider the question of producing concrete examples for all of them. We present the two main methods of construction of such examples and we give several instances of their application. We end the paper outlining the state of the art on this topic and raising several questions.
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Semistable fibrations over an elliptic curve with only one singular fibre
In this work we describe a construction of semistable fibrations over an elliptic curve with one unique singular fibre and we give effective examples using monodromy of curves.
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Godeaux surfaces with an Enriques involution and some stable degenerations
We give an explicit description of the Godeaux surfaces that admit an involution such that the quotient surface is birational to an Enriques surface; these surfaces give a 6-dimensional unirational irreducible subset of the moduli space of surfaces of general type. In addition, we describe the Enriques surfaces that are birational to the quotient of a Godeaux surface by an involution and we show that they give a 5-dimensional unirational irreducible subset of the moduli space of Enriques surfaces. Finally, by degenerating our description we obtain some examples of non-normal stable Godeaux surfaces.
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The Hodge number h^1,1 of irregular algebraic surfaces
We prove a new inequality for the Hodge number h^1,1 of irregular complex smooth projective surfaces of general type without irregular pencils of genus >1. More specifically we show that if the irregularity q satisfies q=2^k+1 then h^1,1>4q-4.
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The classification of minimal irregular surfaces of general type with K^2= 2p_g
Minimal irregular surfaces of general type satisfy K^2\geq 2p_g. In this paper we classify those surfaces for which the equality K^2=2p_g holds.
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Abelian varieties in Brill--Noether loci
In this paper, improving on results of Abramovich- Harris and Debarre- Fahlaoui we give the full classification of curves C of genus g such that a Brill--Noether locus W^ s_d(C), strictly contained in the jacobian J(C) of C, contains a variety Z stable under translations by the elements of a positive dimensional abelian subvariety A contained in J(C) and such that dim(Z)=d-dim(A)-2s, i.e., the maximum possible for such a Z.
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On Schoen surfaces
We give a new construction of the irregular, generalized Lagrangian, surfaces of general type with p_g=5, χ=2, K^2=8, recently discovered by Chad Schoen. Our approach proves that, if S is a general Schoen surface, its canonical map is a finite morphism of degree 2 onto a canonical surface with invariants p_g=5, χ=6, K^2=8, a complete intersection of a quadric and a quartic hypersurface in P^4, with 40 even nodes.
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Brill-Noether loci for divisors on irregular varieties
For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={η\in Pic^0(X)|h^0(L+η)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent space. If X is a smooth surface of maximal Albanese dimension and C is a curve on X, we define a Brill-Noether number ρ(C, r) and we prove, under some mild additional assumptions, that if ρ(C, r) is non negative then W^r(C,X) is nonempty of dimension bigger or equal to ρ(C,r). As an application, we derive lower bounds for h^0(K_D) for a divisor D that moves linearly on a smooth projective variety X of maximal Albanese dimension and inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension.
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Continuous families of divisors, paracanonical systems and a new inequality for varieties of maximal Albanese dimension
Given a smooth complex projective variety X, a line bundle L of X an element v of H^1(O_X) and a section s in H^0(L) that deforms to first order in the direction v, we give a sufficient condition on v in terms of Koszul cohomology for this first order deformation to extend to an analytic deformation. We apply this result to improve known results on the paracanonical system of a variety of maximal Albanese dimension, due to Beauville in the case of surfaces and to Lazarsfeld-Popa in higher dimension. In particular, we prove the inequality p_g(X)>=χ(K_X)+q(X)-1 for a variety X of maximal Albanese dimension without irregular fibrations of Albanese general type.
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A uniform bound on the canonical degree of Albanese defective curves on surfaces
Let S be a minimal complex surface of general type with irregularity q>=2 and let C be an irreducible curve of geometric genus g contained in S. Assume that C is "Albanese defective", i.e., that the image of C via the Albanese map does not generate the Albanese variety Alb(S); we obtain a linear upper bound in terms of K^2_S and g for the canonical degree K_SC of C. As a corollary, we obtain a bound for the canonical degree of curves with g<= q-1, thereby generalizing and sharpening the main result of [S.Y. Lu, On surfaces of general type with maximal Albanese dimension, J. Reine Angew. Math. 641 (2010), 163-175].
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On the canonical map of surfaces with q>=6
We carry out an analysis of the canonical system of a minimal complex surface of general type with irregularity q>0. Using this analysis we are able to sharpen in the case q>0 the well known Castelnuovo inequality K^2>=3p_g+q-7. Then we turn to the study of surfaces with p_g=2q-3 and no fibration onto a curve of genus >1. We prove that for q>=6 the canonical map is birational. Combining this result with the analysis of the canonical system, we also prove the inequality: K^2>=7χ+2. This improves an earlier result of the first and second author [M.Mendes Lopes and R.Pardini, On surfaces with p_g=2q-3, Adv. in Geom. 10 (3) (2010), 549-555].
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