Smooth Irreducible Curve Research Articles

Constructing stable vector bundles from curves with torsion normal bundle

Given a smooth irreducible curve $S$ with torsion normal bundle on a projective surface $X$, we provide a criterion for the non-emptiness of the moduli of slope stable vector bundles with prescribed Chern classes. The criterion is given in terms of the topology of the pair $(X,S)$.

Holomorphic differentials of Klein four covers

Let k be an algebraically closed field of characteristic two, and let G be isomorphic to Z/2×Z/2. Suppose X is a smooth projective irreducible curve over k with a faithful G-action, and assume that the cover X→X/G is totally ramified, in the sense that it is ramified and every branch point is totally ramified. We study to what extent the lower ramification groups of the closed points of X determine the isomorphism types of the indecomposable kG-modules and the multiplicities with which they occur as direct summands of the space H0(X,ΩX/k) of holomorphic differentials of X over k. In the case when X/G=Pk1, we completely determine the decomposition of H0(X,ΩX/k) into a direct sum of indecomposable kG-modules. Moreover, we show that the isomorphism classes of indecomposable kG-modules that actually occur as direct summands belong to an infinite list of non-isomorphic indecomposable kG-modules that contain modules of arbitrarily large k-dimension. In particular, our results show that [14, Theorem 6.4] is incorrect.

Existence and reducibility of the Hilbert scheme of smooth and linearly normal curves in Pr of relatively high degrees

Let H d , g , r be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree d and genus g in P r . We denote by H d , g , r L the union of those components of H d , g , r whose general element is linearly normal. In this article we show that H d , g , r L ( d ≥ g + r − 3 ) is non-empty in a certain optimal range of triples ( d , g , r ) and is empty outside the range. This settles the existence (or non-emptiness if one prefers) of the Hilbert scheme H d , g , r L of linearly normal curves of degree d and genus g in P r for g + r − 3 ≤ d ≤ g + r , r ≥ 3 . We also determine all the triples ( d , g , r ) with g + r − 3 ≤ d ≤ g + r for which H d , g , r L is reducible (or irreducible).

The fibers of the ramified Prym map

We study the ramified Prym map [Formula: see text] which assigns to a ramified double cover of a smooth irreducible curve of genus [Formula: see text] ramified in [Formula: see text] points the Prym variety of the covering. We focus on the six cases where the dimension of the source is strictly greater than the dimension of the target giving a geometric description of the generic fiber. We also give an explicit example of a totally geodesic curve which is an irreducible component of a fiber of the Prym map [Formula: see text].

Unlikely intersections between isogeny orbits and curves

Fix an abelian variety A_0 and a non-isotrivial abelian scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of translates of a fixed finite-rank subgroup of A_0 , also defined over the algebraic numbers, by abelian subvarieties of A_0 of codimension at least k under all isogenies between A_0 and some fiber of the abelian scheme. We characterize the curves inside the abelian scheme which are defined over the algebraic numbers, dominate the base curve and potentially intersect this set in infinitely many points. Our proof follows the Pila–Zannier strategy.

Lubin-Tate extensions and Carlitz module over a projective line: an explicit connection

In this article we consider different approaches for constructing maximal abelian extensions for local and global geometric fields. The Lubin–Tate theory plays key role in the maximal abelian extension construction for local geometric fields. In the case of global geometric fields, Drinfeld modules are of particular interest. In this paper we consider the simpliest special case of Drinfeld modules for projective line which is called the Carlitz module. In the introduction, we provide motivation and a brief historical background on the topics covered in the work. In the first and second sections we provide brief information about Lubin–Tate modules and Carlitz module. In the third section we present two main results: • an explicit connection between the local and global field theory in the geometric case for projective line over finite field: it is proved that the extension tower of Carlitz module induces the tower of the Lubin–Tate extensions. • a connection between Artin maps of extensions of a function field of an arbitrary projective smooth irreducible curve and extensions of completions of local rings at closed points of this curve. In the last section we formulate different open problems and interesting directions for further research, which include generalization first result for an arbitrary smooth projective curve over a finite field and consideration Drinfeld modules of higher rank.

Finite index theorems for iterated Galois groups of unicritical polynomials

Let K K be the function field of a smooth irreducible curve defined over Q ¯ \overline {Q} . Let f ∈ K [ x ] f\in K[x] be of the form f ( x ) = x q + c f(x)=x^q+c , where q = p r , r ≥ 1 , q = p^{r}, r \ge 1, is a power of the prime number p p , and let β ∈ K ¯ \beta \in \overline {K} . For all n ∈ N ∪ { ∞ } n\in \mathbb {N}\cup \{\infty \} , the Galois groups G n ( β ) = G a l ( K ( f − n ( β ) ) / K ( β ) ) G_n(\beta )=\mathrm {Gal}(K(f^{-n}(\beta ))/K(\beta )) embed into [ C q ] n [C_q]^n , the n n -fold wreath product of the cyclic group C q C_q . We show that if f f is not isotrivial, then [ [ C q ] ∞ : G ∞ ( β ) ] > ∞ [[C_q]^\infty :G_\infty (\beta )]>\infty unless β \beta is postcritical or periodic. We are also able to prove that if f 1 ( x ) = x q + c 1 f_1(x)=x^q+c_1 and f 2 ( x ) = x q + c 2 f_2(x)=x^q+c_2 are two such distinct polynomials, then the fields ⋃ n = 1 ∞ K ( f 1 − n ( β ) ) \bigcup _{n=1}^\infty K(f_1^{-n}(\beta )) and ⋃ n = 1 ∞ K ( f 2 − n ( β ) ) \bigcup _{n=1}^\infty K(f_2^{-n}(\beta )) are disjoint over a finite extension of K K .

On the Hilbert scheme of smooth curves in ℙ4 of degree d = g + 1 and genus g with negative Brill–Noether number

We denote by [Formula: see text] the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree [Formula: see text] and genus [Formula: see text] in [Formula: see text]. In this paper, we show that for low genus [Formula: see text] outside the Brill–Noether range, the Hilbert scheme [Formula: see text] is non-empty whenever [Formula: see text] and irreducible whose only component generically consists of linearly normal curves unless [Formula: see text] or [Formula: see text]. This complements the validity of the original assertion of Severi regarding the irreducibility of [Formula: see text] outside the Brill–Nother range for [Formula: see text] and [Formula: see text].

A new series of moduli components of rank-2 semistable sheaves on with singularities of mixed dimension

We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme , , of semistable rank-2 sheaves on with Chern classes , and , whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly -semistable reflexive sheaves along disjoint unions of collections of points and smooth irreducible curves which are rational or complete intersection curves in . As a special member of this series, we obtain a new component of . Bibliography: 12 titles.

Unlikely intersections with isogeny orbits in a product of elliptic schemes

Fix an elliptic curve $E_0$ without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of $E_0^g$, also defined over the algebraic numbers, under all isogenies between $E_0^g$ and some fiber of the $g$-th fibered power $\mathcal{A}$ of the elliptic scheme, where $g$ is a fixed natural number. As a special case of a slightly more general result, we characterize the subvarieties (of arbitrary dimension) inside $\mathcal{A}$ that have potentially Zariski dense intersection with this set. In the proof, we combine a generalized Vojta-R\'emond inequality with the Pila-Zannier strategy.

Components of the Hilbert scheme of smooth projective curves using ruled surfaces

Let $${\mathcal {I}}_{d,g,r}$$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree d and genus g in $$\mathbb {P}^r$$ . We use families of curves on cones to show that under certain numerical assumptions for d, g and r, the scheme $${\mathcal {I}}_{d,g,r}$$ acquires generically smooth components whose general points correspond to curves that are double covers of irrational curves. In particular, in the case $$\rho (d,g,r) := g-(r+1)(g-d+r) \ge 0$$ we construct explicitly a regular component that is different from the distinguished component of $${\mathcal {I}}_{d,g,r}$$ dominating the moduli space $${\mathcal {M}}_g$$ . Our result implies also that if $$g \ge 57$$ then $${\mathcal {I}}_{\frac{4g}{3}, g, \frac{g+1}{2}}$$ has at least two generically smooth components parametrizing linearly normal curves.

Unlikely intersections in families of abelian varieties and the polynomial Pell equation

Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that when A is a fibered product of elliptic schemes, if C is not contained in a proper subgroup scheme of A, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g bigger or equal than 2. This, combined with the above mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes. These results have applications in the study of solvability of almost-Pell equations in polynomials.

On the Hilbert scheme of linearly normal curves in [formula omitted] of relatively high degree

Let Hd,g,r be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree d and genus g in Pr. We denote by Hd,g,rL the union of those components of Hd,g,r whose general element is linearly normal and we show that any non-empty Hd,g,rL (d≥g+r−3) is irreducible for an extensive range of triples (d,g,r) beyond the Brill-Noether range. This establishes the validity of a suitably modified assertion of Severi regarding the irreducibility of the Hilbert scheme Hd,g,rL of linearly normal curves for g+r−3≤d≤g+r, r≥3, and g≥2r+3 if d=g+r−3.

On the Hilbert scheme of linearly normal curves in $$\mathbb {P}^4$$ of degree $$d = g+1$$ and genus g

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this article, we show that any non-empty $\mathcal{H}_{g+1,g,4}$ has only one component whose general element is linear normal unless $g=9$. If $g=9$, we show that $\mathcal{H}_{g+1,g,4}$ is reducible with two components and a general element of each component is linearly normal. This establishes the validity of a certain modified version of an assertion of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$ for the case $d=g+1$ and $r=4$.

Nef and pseudoeffective cones of product of projective bundles over a curve

Let X=P(E1)×CP(E2), where C is any smooth irreducible curve and E1, E2 are vector bundles over C. In this paper, we calculate the nef cone and pseudoeffective cone of X whenever E1 and E2 are semistable, and show that they coincide. We also calculate the nef cone and pseudoeffective cone of X in case both E1 and E2 have rank 2, and neither E1 nor E2 is semistable or at least one of them is semistable. As a result, in the case when both E1 and E2 are of rank 2 bundles on C, we get that nef cone and pseudoeffective cone of X coincide iff both E1 and E2 are semistable.

Irreducibility and components rigid in moduli of the Hilbert scheme of smooth curves

Denote by $$\mathcal {H}_{d,g,r}$$ the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in $$\mathbb {P}^r$$ . A component of $$\mathcal {H}_{d,g,r}$$ is rigid in moduli if its image under the natural map $$\pi :\mathcal {H}_{d,g,r} \dashrightarrow \mathcal {M}_{g}$$ is a one point set. In this note, we provide a proof of the fact that $$\mathcal {H}_{d,g,r}$$ has no components rigid in moduli for $$g > 0$$ and $$r=3$$ , from which it follows that the only smooth projective curves embedded in $$\mathbb {P}^3$$ whose only deformations are given by projective transformations are the twisted cubic curves. In case $$r \ge 4$$ , we also prove the non-existence of a component of $$\mathcal {H}_{d,g,r}$$ rigid in moduli in a certain restricted range of d, $$g>0$$ and r. In the course of the proofs, we establish the irreducibility of $$\mathcal {H}_{d,g,3}$$ beyond the range which has been known before.

On the Hyperbolicity Locus of a Real Curve

Given a real algebraic curve in the projective 3-space, its hyperbolicity locus is the set of lines with respect to which the curve is hyperbolic. We give an example of a smooth irreducible curve whose hyperbolicity locus is disconnected but the connected components are not distinguished by the linking numbers with the connected components of the curve.

The Gauss–Manin connection on the periodic cyclic homology

Let R be the algebra of functions on a smooth affine irreducible curve S over a field k and let $${A_{\bullet }}$$ be a smooth and proper DG algebra over R. The relative periodic cyclic homology $$HP_* ({A_{\bullet }})$$ of $${A_{\bullet }}$$ over R is equipped with the Hodge filtration $${\mathcal F}^{\cdot }$$ and the Gauss–Manin connection $$\nabla $$ (Getzler, in: Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), Israel mathematics conference proceedings, vol 7, Bar-Ilan University, Ramat Gan, pp 65–78, 1993; Kaledin, in: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, vol II, pp 23–47, Progress in mathematics, vol 270, Birkhauser Inc., Boston, 2009) satisfying the Griffiths transversality condition. When k is a perfect field of odd characteristic p, we prove that, if the relative Hochschild homology $$HH_m({A_{\bullet }}, {A_{\bullet }})$$ vanishes in degrees $$|m| \ge p-2$$ , then a lifting $$\tilde{R}$$ of R over $$W_2(k)$$ and a lifting of $${A_{\bullet }}$$ over $$\tilde{R}$$ determine the structure of a relative Fontaine–Laffaille module (Faltings, in: Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins University Press, Baltimore, MD, pp 25–80, 1989, §2 (c); Ogus and Vologodsky in Publ Math Inst Hautes Etudes Sci No 106:1–138, 2007 §4.6) on $$HP_* ({A_{\bullet }})$$ . That is, the inverse Cartier transform of the Higgs R-module $$(Gr^{\mathcal F}HP_* ({A_{\bullet }}), Gr^{\mathcal F}\nabla )$$ is canonically isomorphic to $$ (HP_* ({A_{\bullet }}), \nabla )$$ . This is non-commutative counterpart of Faltings’ result (1989, Th. 6.2) for the de Rham cohomology of a smooth proper scheme over R. Our result amplifies the non-commutative Deligne–Illusie decomposition proven by Kaledin (Algebra, geometry and physics in the 21st century (Kontsevich Festschrift), Progress in mathematics, vol 324. Birkhauser, pp 99–129, 2017, Th. 5.1). As a corollary, we show that the p-curvature of the Gauss–Manin connection on $$HP_* ({A_{\bullet }})$$ is nilpotent and, moreover, it can be expressed in terms of the Kodaira–Spencer class $$\kappa \in HH^2({A_{\bullet }}, {A_{\bullet }}) \otimes _R \Omega ^1_R$$ [a similar result for the p-curvature of the Gauss–Manin connection on the de Rham cohomology is proven by Katz (Invent Math 18:1–118, 1972)]. As an application of the nilpotency of the p-curvature we prove, using a result from Katz (Inst Hautes Etudes Sci Publ Math No 39:175–232, 1970), a version of “the local monodromy theorem” of Griffiths–Landman–Grothendieck for the periodic cyclic homology: if $$k={\mathbb C}$$ , $$\overline{S}$$ is a smooth compactification of S, then, for any smooth and proper DG algebra $${A_{\bullet }}$$ over R, the Gauss–Manin connection on the relative periodic cyclic homology $$HP_* ({A_{\bullet }})$$ has regular singularities, and its monodromy around every point at $$\overline{S}-S$$ is quasi-unipotent.

Projectively normal embeddings of an algebraic curve with a net

Let X be a smooth irreducible algebraic curve of genus g. The projective normality of a complete embedding of X is determined by only its quadratic normality in case the embedding is of degree at least g. This means that the complete embedding fails to be projectively normal if and only if it admits an effective divisor which fails to impose independent conditions on quadrics in the embedded projective space. Thus if X admits a net, then it is interesting to compare the conditions for the projective normality of an embedding of X with properties of conic sections of the plane curve given by the net. For such a curve X with a net, we show that the projectively normal embeddings are closely related to properties of conic sections.

On curves on K3 surfaces, II

We study the Clifford index c of a smooth irreducible curve X in the linear series |2H| on a special K3 surface S of degree 2n in $${{\mathbb P}}^{n+1}$$ , with hyperplane section H, and we look for the complete and base point free linear series of S whose restrictions to X compute c. In a more general context, we discuss the features of such series, for an assigned curve on a K3 surface; this discussion is of some independent interest.