Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois coverings.
We describe various properties of Hirzebruch surfaces and related constructions: degenerations, braid monodromy, Galois covers and their Chern classes.
In this paper we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based …
We consider, under suitable assumptions, the following situation: $\mathcal B$ is a component of the moduli space of polarized surfaces and $\mathcal V_{m,δ}$ is the universal Severi variety over $\mathcal B$ parametrizing pairs $(S,C)$, with $(S,H)\in \mathcal B$ and $C\in |mH|$ irreducible with exactly $δ$ nodes as singularities. The moduli map $\mathcal V\to \mathcal M_g$ of an irreducible component $\mathcal V$ of $\mathcal V_{m,δ}$ is generically of maximal rank if and only if certain cohomology vanishings hold. Assuming there are suitable semistable degenerations of the surfaces in $\mat…
Over a family $\mathbb X$ of genus $g$ complete curves, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of ${\rm SL}(n, \mathbb C)$ bundles over the generic smooth curve $X_t$ in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by $(\mathbb C^*)^s$ of a moduli space on the desingularisation. Taking a "maximal" degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we …
In this note we study the geometry of torsors under flat and finite commutative group schemes of rank p above curves in characteristic p and above relative curves over a complete discrete valuation ring of inequal characteristics. In bothe cases we study the Galois action of the Galois group of the base field on these torsors. We also study degeneration of $μ_p$-torsors from characteristic 0 to characteristic p and show that this degeneration is compatible with the Galois action. We then discuss the lifting of torsors under flat and commutative group schemes of rank p from positive to 0 charac…
In this paper we answer two long-standing questions in the classification of $G$-torsors on curves for an almost simple, simply connected algebraic group $G$ over the field of complex numbers. The first question is to give an intrinsic definition of (semi)stability for a $G$-torsor on an {\em irreducible nodal curve} and the second one is the construction of a flat degeneration of the moduli space of semistable $G$-torsors when the smooth curve degenerates to an irreducible nodal curve. A generalization of the classical Bruhat-Tits group schemes to two-dimensional regular local rings and an ap…
Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of constructing and studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of $δ$-semistable pseudo bundles on a nodal curve constructed by the first author become, for large values of $δ$, the moduli spaces for semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis of further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundle…
For a semi simple group G it is known the moduli stack of principal G-bundles over a fixed nodal curve is not complete. Finding a completion requires compactifying the group G. However it was shown in [34] that this is not sufficient to complete the moduli stack over a family of curves. In this paper I describe how to use an embedding of the loop group LG to provide a completion of the stack of G-bundles over a one dimensional family of curves degenerating to a nodal curve. The completion comes with a modular interpretation inspired by the work of Gieseker, Seshadri, Kausz and Thaddeus and Mar…
Let T be a complex torus, and X the surface CP^1 x T. If T is embedded in CP^{n-1} then X may be embedded in CP^{2n-1}. Let X_Gal be its Galois cover with respect to a generic projection to CP^2. In this paper we compute the fundamental group of X_Gal, using the degeneration and regeneration techniques, the Moishezon-Teicher braid monodromy algorithm and group calculations. We show that pi_1(X_Gal) = Z^{4n-2}.
In this note, we revisit the modified diagonal cycle of Gross and Schoen. We look at degenerations of this cycle, induced by a degeneration of the curve C, and explain how the specialization map with respect to the central fiber produces a higher Chow cycle. When C is non-hyperelliptic of genus three, and degenerates to a nodal curve, the degeneration of the cycle corresponds to an indecomposable higher Chow cycle on a curve of genus two.
The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal crossing singularity and it provides a flat degeneration. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than $1$) and fixed degree such that rank and degree are co-prime.
Mumford and Newstead generalized the classical Torelli theorem to higher rank i.e., a smooth, projective curve $X$ is uniquely determined by the second intermediate Jacobian of the moduli space of stable rank $2$ bundles on $X$, with fixed odd degree determinant. In this article we prove the analogous result in the case $X$ is an irreducible nodal curve with one node. As a byproduct, we obtain the degeneration of the second intermediate Jacobians and the associated Néron model of a family of such moduli spaces.
Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smoothability criteria. As applications, we show that the spin parity of holomorphic and meromorphic differentials extends to distinguish twisted canonical divisors in the locus of stable pointed curves of pseudocompact type. We also justify whether zeros and poles on general curves in …
This paper aims at settling the issue of the validity of the de Jonquières formulas. We consider the space of divisors with prescribed multiplicity, or de Jonquières divisors, contained in a linear series on a smooth projective curve. Assuming zero expected dimension of this space, the de Jonquières formulas compute the virtual number of de Jonquières divisors. Using degenerations to nodal curves we show that for a general curve equipped with a general complete linear series, the space is of expected dimension, which shows that the counts are in fact true. This implies that in the case of nega…
Let T be the complex projective torus, and X the surface CP^1 X T. Let X_Gal be its Galois cover with respect to a generic projection to CP^2. In this paper we compute the fundamental group of X_Gal, using the degeneration and regeneration techniques, the Moishezon-Teicher braid monodromy algorithm and group calculations. We show that pi_1(X_Gal) = Z^10.
We study the construction of complex minimal smooth surfaces $S$ of general type with $p_g(S)=0$ and $K_S^2=7$. Inoue constructed the first examples of such surfaces, which can be described as Galois $\mathbb{Z}_2\times\mathbb{Z}_2$-covers over the four-nodal cubic surface. Later the first named author constructed more examples as Galois $\mathbb{Z}_2\times\mathbb{Z}_2$-covers over certain six-nodal del Pezzo surfaces of degree one. In this paper we construct a two-dimensional family of minimal smooth surfaces of general type with $p_g=0$ and $K^2=7$, as Galois $\mathbb{Z}_2\times\mathbb{Z}_2$…
Let $X$ be the surface $\T\times\T$ where $\T$ is the complex torus. This paper is the third in a series, studying the fundamental group of the Galois cover of $X$ \wrt a generic projection onto $\C¶^2$. Van Kampen Theorem gives a presentation of the fundamental group of the complement of the branch curve, with 54 generators and more than 2000 relations. Here we introduce a certain natural quotient (obtained by identifying pairs of generators), prove it is a quotient of a Coxeter group related to the degeneration of $X$, and show that this quotient is virtually nilpotent.
We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear series with base points over families of nodal curves of compact type, which we then use to compute combinatorial formulas for the number of secant-exceptional linear series when the spaces of linear…
In this paper, we develop the theory of flashes of an algebraic curve. We show that the theory is birationally invariant in a sense which we will make more precise below. We also show how the theory provides a foundation for the method of asymptotic degenerations, a particular class of degenerations of plane projective algebraic curves. In particular, we consider the geometrical technique in relation to the Severi problem of degenerating nodal curves to lines in general position, and correct Severi's original proof of his conjecture.
Newstead gave the generators of the cohomology ring of the moduli space of rank 2 semi-stable, torsion-free sheaves with fixed odd degree determinant over a smooth, projective curve. In this article, we generalize this result to the case when the underlying curve is irreducible, nodal. We show that these generators (of the cohomology ring in the nodal curve case) arise naturally as degeneration of Newstead's generators in the smooth curve case.
This paper is the second in a series. The first one describes pillow degenerations of a $K3$ surface with genus $g$. In this paper we study the $(2,2)$-pillow degeneration of a non-prime $K3$ surface and the braid monodromy of the branch curve of the surface with respect to a generic projection onto $\C¶^2$. In future papers we study the fundamental group of the complement of the branch curve and the fundamental group of the Galois cover of the surface with respect to this generic projection.
We address the following question: Given a polarized toric surface (S,L), and a general integral curve C of geometric genus g in the linear system |L|, do there exist degenerations of C in |L| to general integral curves of smaller geometric genera? We give an affirmative answer to this question for surfaces associated to h-transverse polygons, provided that the characteristic of the ground field is large enough. We give examples of surfaces in small characteristic, for which the answer to the question is negative. In case the answer is affirmative, we deduce that a general curve C as above is …
We argue that for a smooth surface S, considered as a ramified cover over the projective plane branched over a nodal-cuspidal curve B one could use the structure of the fundamental group of the complement of the branch curve to understand other properties of the surface and its degeneration and vice-versa. In this paper, we look at embedded-degeneratable surfaces - a class of surfaces admitting a planar degeneration with a few combinatorial conditions imposed on its degeneration. We close a conjecture of Teicher on the virtual solvability of the mentioned fundamental group for these surfaces a…
Let S be a complex smooth projective surface and L be a line bundle on S. Göttsche conjectured that for every integer r, the number of r-nodal curves in |L| is a universal polynomial of four topological numbers when L is sufficiently ample. We prove Göttsche's conjecture using the algebraic cobordism group of line bundles on surfaces and degeneration of Hilbert schemes of points. In addition, we prove the the Göttsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasi-modular forms and two unknown series.
We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the classical case, this gives a geometrical interpretation of Libgober's divisibility Theorem. We calculate twisted polynomials for some algebraic curves and show how they can detect Zariski pairs of equivalent Alexander polynomials and that they are sensitive to nodal degeneration…
A conjecture of Mumford predicts a complete set of relations between the generators of the cohomology ring of the moduli space of rank 2 semi-stable sheaves with fixed odd degree determinant on a smooth, projective curve of genus at least 2. The conjecture was proven by Kirwan. In this article, we generalize the conjecture to the case when the underlying curve is irreducible, nodal. In fact, we show that these relations (in the nodal curve case) arise naturally as degeneration of the Mumford relations shown by Kirwan in the smooth curve case. As a byproduct, we compute the Hodge-Poincare polyn…
Augmentations, reduced ideal point gluings and compact type degenerations of curves
University of Georgia（佐治亚大学）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；NRU Higher School of Economics（国立研究大学高等经济学院）
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This article and the first two that preceded it are the first in a series aimed to explore this new approach. In Part I, we set up the combinatorial framework and showed how graphs weighted with integer lengths associated to the edges provide tilings of Euclidean spaces by certain polytopes associat…
We describe, for various degenerations $S\to Δ$ of quartic $K3$ surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits as $t\in Δ^*$ tends to 0 of the Severi varieties $V_δ(S_t)$, parametrizing irreducible $δ$-nodal plane sections of $S_t$. We give applications of this to (i) the counting of plane nodal curves through base points in special position, (ii) the irreducibility of Severi varieties of a general quartic surface, and (iii) the monodromy of the universal family of rational curves on quartic $K3$ surfaces.
We study the behavior of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We determine all possible limits of RN differentials in degenerating sequences of smooth curves, and describe the limit in terms of solutions of the corresponding Kirchhoff problem. We further show that the limit of zeroes of RN differentials is the set of zeroes of a twisted meromorphic RN differential, which we explicitly construct. Our main new tool is an explicit solution of the jump problem on Riemann surfaces in plumbing coordinates, by using the Cauchy kernel on the normalizat…
In this paper, we compute the number of covers of curves with given branch behavior in characteristic p for one class of examples with four branch points and degree p. Our techniques involve related computations in the case of three branch points, and allow us to conclude in many cases that for a particular choice of degeneration, all the covers we consider degenerate to separable (admissible) covers. Starting from a good understanding of the complex case, the proof is centered on the theory of stable reduction of Galois covers.
