Degenerations of bundle moduli Biswas, Indranil; Hurtubise, Jacques
Advances in mathematics (New York. 1965),
02/2023, Volume:
414
Journal Article
Peer reviewed
Over a family X of complete curves of genus g, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of holomorphic ...SL(n,C) bundles over the generic smooth curve Xt 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 (C⁎)s of a moduli space on the desingularization. Taking a “maximal” degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we obtain a degeneration of the moduli space of bundles into a (C⁎)(3g−3)(n−1)-quotient of a (2g−2)-th power of a space associated to the three-pointed sphere. Via the Narasimhan-Seshadri theorem, the moduli space of SL(n,C) bundles on the smooth curve is a space of representations of the fundamental group into SU(n) (the “symplectic picture”). We obtain the degenerations also in this symplectic context, in a way that is compatible with the holomorphic degeneration, so that our limit space is also a (S1)(3g−3)(n−1) symplectic quotient of a (2g−2)-th power of a space associated to the three-pointed sphere.
Let X be a (projective, geometrically irreducible, non-singular) algebraic curve of genus g≥2 defined over an algebraically closed field K of odd characteristic p. Let Aut(X) be the group of all ...automorphisms of X which fix K element-wise. It is known that if |Aut(X)|≥8g3 then the p-rank (equivalently, the Hasse–Witt invariant) of X is zero. This raises the problem of determining the (minimum-value) function f(g) such that whenever |Aut(X)|≥f(g) then X has zero p-rank. For eveng we prove that f(g)≤900g2. The odd genus case appears to be much more difficult although, for any genus g≥2, if Aut(X) has a solvable subgroup G such that |G|>252g2 then X has zero p-rank and G fixes a point of X. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from Group theory characterizing finite simple groups whose Sylow 2-subgroups have a cyclic subgroup of index 2. We also point out some connections with the Abhyankar conjecture and the Katz–Gabber covers.
We show how a type of multi-Frobenius nonclassicality of a curve defined over a finite field Fq of characteristic p reflects on the geometry of its strict dual curve. In particular, in such cases we ...may describe all the possible intersection multiplicities of its strict dual curve with the linear system of hyperplanes. Among other consequence, using a result by Homma, we are able to construct nonreflexive space curves such that their tangent surfaces are nonreflexive as well, and the image of a generic point by a Frobenius map is in its osculating hyperplane. We also obtain generalizations and improvements of some known results of the literature.
We study the complexity of multiplication of two elements in a finite field extension given by their coordinates in a normal basis. We show how to control this complexity using the arithmetic and ...geometry of algebraic curves.
Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, ...geometric constructions in computer aided design, image detection, algebraic differential equations, etc.), there often appear unknown parameters. It is possible to adjoin these parameters to the coefficient field as transcendental elements. In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way. In this paper, we show when the singularities and thus the (geometric) genus of the curves might change. More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant. In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible. For the genus zero case, and for a given rational parametrization, a finer partition is possible such that the specialization of the parametrization parametrizes the specialized curve. Moreover, in this case, the set of parameters where Hilbert's irreducibility theorem does not hold can be identified. We conclude the paper by illustrating these results by some concrete applications.
We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all ...realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szegő polynomials. Our results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions. This paper contains new results, some old results presented from a different perspective or with a different proof, and a formal foundation for our analysis. We give a rigorous definition of the Poncelet property, of curves tangent to a family of polygons, and of polygons associated with Poncelet curves. As a result, we are able to clarify some misconceptions that appear in the literature and present counterexamples to some existing assertions along with necessary modifications to their hypotheses to validate them. For instance, we show that curves inscribed in some families of polygons supported on T are not necessarily convex, can have cusps, and can even intersect the unit circle. Two ideas play a unifying role in this work. The first is the utility of OPUC and the second is the advantage of working with tangent coordinates. This latter idea has been previously exploited in the works of B. Mirman, whose contribution we have tried to put in perspective.
Let G be a subgroup of the three dimensional projective group PGL(3,q) defined over a finite field Fq of order q, viewed as a subgroup of PGL(3,K) where K is an algebraic closure of Fq. For ...G≅PGL(3,q) and for the seven nonsporadic, maximal subgroups G of PGL(3,q), we investigate the (projective, irreducible) plane curves defined over K that are left invariant by G. For each, we compute the minimum degree d(G) of G-invariant curves, provide a classification of all G-invariant curves of degree d(G), and determine the first gap ε(G) in the spectrum of the degrees of all G-invariant curves. We show that the curves of degree d(G) belong to a pencil depending on G, unless they are uniquely determined by G. For most examples of plane curves left invariant by a large subgroup of PGL(3,q), the whole automorphism group of the curve is linear, i.e., a subgroup of PGL(3,K). Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.
This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years ...and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves.
The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.
We study the Hilbert scheme Hd,g,rL parametrizing smooth, irreducible, non-degenerate and linearly normal curves of degree d and genus g in Pr whose complete and very ample hyperplane linear series D ...have relatively small index of speciality i(D)=g−d+r. In particular we show the existence (and non-existence as well in some sporadic cases) of every Hilbert scheme of linearly normal curves with i(D)=4. We also determine the irreducibility of H2r+4,r+8,rL for 3≤r≤8, which are rather peculiar families in a certain sense.
The purpose of this paper is to show that for a complete intersection curve C in projective space (other than a few exceptions stated below), any morphism f:C→Pr satisfying degf⁎OPr(1)<degC is ...obtained by projection from a linear space. In particular, we obtain bounds on the gonality of such curves and compute the gonality of general complete intersection curves. We also prove a special case of one of the well-known Cayley-Bacharach conjectures posed by Eisenbud, Green, and Harris.
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