Zariski Open Subset Research Articles

Geometry of certain foliations on the complex projective plane

Let $d\geq2$ be an integer. The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $d^2+4d+2$ on which $\mathrm{Aut}(\mathbb P^2_{\mathbb C})$ acts. We show that there are exactly two orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ of minimal dimension $6$, necessarily closed in $\mathbf{F}(d)$. This generalizes known results in degrees $2$ and $3.$ We deduce that an orbit $\mathcal{O}(\mathcal{F})$ of an element $\mathcal{F}\in\mathbf{F}(d)$ of dimension $7$ is closed in $\mathbf{F}(d)$ if and only if $\mathcal{F}_{i}^{d}\not\in\overline{\mathcal{O}(\mathcal{F})}$ for $i=1,2.$ This allows us to show that in any degree $d\geq3$ there are closed orbits in $\mathbf F(d)$ other than the orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d}),$ unlike the situation in degree $2.$ On the other hand, we introduce the notion of the basin of attraction $\mathbf{B}(\mathcal{F})$ of a foliation $\mathcal{F}\in\mathbf{F}(d)$ as the set of $\mathcal{G}\in\mathbf{F}(d)$ such that $\mathcal{F}\in\overline{\mathcal{O}(\mathcal{G})}.$ We show that the basin of attraction $\mathbf{B}(\mathcal{F}_{1}^{d})$, resp. $\mathbf{B}(\mathcal{F}_{2}^{d})$, contains a quasi-projective subvariety of $\mathbf{F}(d)$ of dimension greater than or equal to $\dim\mathbf{F}(d)-(d-1)$, resp. $\dim \mathbf{F}(d)-(d-3)$. In particular, we obtain that the basin $\mathbf{B}(\mathcal{F}_{2}^{3})$ contains a non-empty Zariski open subset of $\mathbf{F}(3)$. This is an analog in degree $3$ of a result on foliations of degree $2$ due to Cerveau, D\'eserti, Garba Belko and Meziani.

Seiberg–Witten differentials on the Hitchin base

In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss–Manin) derivative of the Seiberg–Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration.

Stability in the category of smooth mod-p representations of

Abstract Let $p \geq 5$ be a prime number, and let $G = {\mathrm {SL}}_2(\mathbb {Q}_p)$ . Let $\Xi = {\mathrm {Spec}}(Z)$ denote the spectrum of the centre Z of the pro-p Iwahori–Hecke algebra of G with coefficients in a field k of characteristic p. Let $\mathcal {R} \subset \Xi \times \Xi $ denote the support of the pro-p Iwahori ${\mathrm {Ext}}$ -algebra of G, viewed as a $(Z,Z)$ -bimodule. We show that the locally ringed space $\Xi /\mathcal {R}$ is a projective algebraic curve over ${\mathrm {Spec}}(k)$ with two connected components and that each connected component is a chain of projective lines. For each Zariski open subset U of $\Xi /\mathcal {R}$ , we construct a stable localising subcategory $\mathcal {L}_U$ of the category of smooth k-linear representations of G.

Approximation and homotopy in regulous geometry

Let $X$ , $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$ -regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$ , we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$ -regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$ -regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$ -dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$ -regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$ .

Surjective morphisms from affine space to its Zariski open subsets

We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set [Formula: see text], we construct an endomorphism of [Formula: see text] with [Formula: see text] as its image. By Noether’s normalization lemma, these results extend to give surjective maps from any [Formula: see text]-dimensional affine variety [Formula: see text] to [Formula: see text].

Height coincidences in products of the projective line

We consider hypersurfaces in $${\mathbb {P}}_1^n$$ that contain a generic sequence of small dynamical height with respect to a split map and project onto $$n-1$$ coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface $$n-1$$ coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps.

Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics

We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form q(x1,⋯,xn)=m, where q is a non-degenerate integral quadratic form in n⩾3 variables and m is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.

Hyperbolicity for Log Smooth Families with Maximal Variation

Abstract We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation.

Logarithmic Connections, WZNW Action, and Moduli of Parabolic Bundles on the Sphere

Moduli spaces of stable parabolic bundles of parabolic degree 0 over the Riemann sphere are stratified according to the Harder–Narasimhan filtration of underlying vector bundles. Over a Zariski open subset mathscr {N}_{0} of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function mathscr {S} is defined as the regularized critical value of the non-compact Wess–Zumino–Novikov–Witten action functional. The definition of mathscr {S} depends on a suitable notion of parabolic bundle ‘uniformization map’ following from the Mehta–Seshadri and Birkhoff–Grothendieck theorems. It is shown that -mathscr {S} is a primitive for a (1,0)-form vartheta on mathscr {N}_{0} associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that -mathscr {S} is a Kähler potential for (Omega -Omega _{mathrm {T}})|_{mathscr {N}_{0}}, where Omega is the Narasimhan–Atiyah–Bott Kähler form in mathscr {N} and Omega _{mathrm {T}} is a certain linear combination of tautological (1, 1)-forms associated with the marked points. These results provide an explicit relation between the cohomology class [Omega ] and tautological classes, which holds globally over certain open chambers of parabolic weights where mathscr {N}_{0} = mathscr {N}.

Deformation quantization of moduli spaces of Higgs bundles on a Riemann surface with translation structure

Let X be a compact connected Riemann surface of genus g ≥ 1 equipped with a nonzero holomorphic 1-form. Let MX(r) denote the moduli space of semistable Higgs bundles on X of rank r and degree r(g − 1) + 1; it is a complex symplectic manifold. Using the translation structure on the open subset of X where the 1-form does not vanish, we construct a natural deformation quantization of a certain nonempty Zariski open subset of MX(r).

The density of rational lines on hypersurfaces: a bihomogeneous perspective

Let F be a non-singular homogeneous polynomial of degree d in n variables. We give an asymptotic formula of the pairs of integer points (mathbf {x}, mathbf {y}) with |mathbf {x}| leqslant X and |mathbf {y}| leqslant Y which generate a line lying in the hypersurface defined by F, provided that n > 2^{d-1}d^4(d+1)(d+2). In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of X and Y.

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\cat{M}_{lc}(n,d)$ denote the moduli space of pairs $(E,D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic connection on $E$ singular over $S$, with fixed residues in the centre of $\mathfrak{gl}(n,\C)$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $\cat{M}'_{lc}(n,d)$ and $\cat{M}_{lc}(n,L)$ be the moduli spaces parametrising all pairs $(E,D)$ such that underlying vector bundle $E$ is stable and $(\bigwedge^nE, \tilde{D}) \cong (L,D_L)$ respectively. Let $\cat{M}'_{lc}(n,L) \subset \cat{M}_{lc}(n,L)$ be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of $\cat{M}'_{lc}(n,d)$ and $\cat{M}'_{lc}(n,L)$ and compute their Picard groups. We also show that $\cat{M}'_{lc}(n,L)$ and hence $\cat{M}_{lc}(n,L)$ do not have any non-constant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over $S$, with arbitrary residues.

ALGEBRAIC FIBER SPACES AND CURVATURE OF HIGHER DIRECT IMAGES

AbstractLet$p:X\rightarrow Y$be an algebraic fiber space, and let$L$be a line bundle on$X$. In this article, we obtain a curvature formula for the higher direct images of$\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$restricted to a suitable Zariski open subset of$X$. Our results are particularly meaningful if$L$is semi-negatively curved on$X$and strictly negative or trivial on smooth fibers of$p$. Several applications are obtained, including a new proof of a result by Viehweg–Zuo in the context of a canonically polarized family of maximal variation and its version for Calabi–Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers – and the complications that are induced by them.

Approximation by piecewise-regular maps

A real algebraic variety W of dimension m is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of Rm. Let l be any nonnegative integer. We prove that every map of class Cl from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the Cl topology by piecewise-regular maps of class Ck, where k is an arbitrary integer satisfying k≥l. Next we derive consequences regarding algebraization of topological vector bundles.

Cayley–Bacharach and Singularities of Foliations

This paper deals with foliations by curves [s] of degree $$ r \ge 2 $$ on $$ \mathbb {P}^2 $$ with isolated singularities S, called non-degenerate if S is reduced and otherwise degenerate. Say that [s] is uniquely determined by a zero-dimensional $$ Y \subset S $$ if [s] is the unique foliation that vanishes on Y and say that Y is minimal for [s] if, moreover, the degree of Y is the minimal possible to do so. Previous work of the authors show that every non-degenerate foliation in degrees $$ 2 \le r \le 5 $$ does have a minimal subscheme and that the set of non-degenerate foliations of degree $$ r \ge 6 $$ that have a minimal subscheme contains a Zariski-open subset of the space of foliations of degree r. For non-degenerate foliations [s] we present both characterizations and sufficient conditions for [s] to have a minimal subscheme. We also give examples of degenerate foliations of degree $$ r \ge 7 $$ that do not have a minimal subscheme at all.

Relative Non-cuspidality of Representations Induced from Split Parabolic Subgroups

Let $G/H$ be a reductive symmetric space over a $p$-adic field, defined by an involution $\sigma$ on $G$. We study the induced representations ${\rm Ind}_P^G(\rho)$ for any proper $\sigma$-split parabolic subgroup $P=MU$ and any $M\cap H$-distinguished representation $\rho$ of $M$. Let ${\mathcal X}$ denote the complex torus of unramified characters of $M$ which are trivial on $M\cap H$ and $\rho_{\chi}$ the twist of $\rho$ by $\chi\in {\mathcal X}$. We show that ${\rm Ind}_P^G(\rho_{\chi})$ is not $H$-relatively cuspidal {\it for generic} $\chi$, that is, for $\chi$ in a Zariski-open subset of ${\mathcal X}$.

Counting rational points on biprojective hypersurfaces of bidegree (1,2)

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biprojective hypersurfaces of bidegree (1,2) in sufficiently many variables. This confirms the Manin conjecture for this variety.

Finite $\mathcal{A}$-determinacy of generic homogeneous map germs in $\mathbb{C}^3$

Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \mathbb{C}^3 \to \mathbb{C}^3$, such that $\deg f_i=d_i$. We show that if $\gcd(d_i,d_j) \leq 2$ for $1 \leq i < j\leq 3$ and $\gcd(d_1,d_2,d_3)=1$, then there is a non-empty Zariski open subset $U \subset H(d_1,d_2,d_3)$ such that for every mapping $F\in U$ the map germ $(F,0)$ is $\mathcal{A}$-finitely determined. Moreover, in this case we compute the number of discrete singularities (0-stable singularities) of a generic mapping $(f_1,f_2,f_3):\mathbb{C}^3 \to \mathbb{C}^3$, where $\deg f_i=d_i$.

Cohomology of the space of polynomial maps on [formula omitted] with prescribed ramification

In this paper we study the moduli spaces Simpnm of degree n+1 morphisms AK1→AK1 with “ramification length <m” over an algebraically closed field K, where we introduce the notion of the ramification length of a morphism to quantify the complexity in its ramification behavior. For each m, the moduli space Simpnm is a Zariski open subset of the space of degree n+1 polynomials over K up to Aut(AK1). It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes- here we are prescribing, instead, the ramification data. Exploiting the topological properties of the poset that encodes the ramification behavior, we use a sheaf-theoretic argument to compute H⁎(Simpnm(C);Q) as well as the étale cohomology He´t⁎(Simpnm/K;Qℓ) for charK=0 or charK>n+1, when n and m are such that n≥3m. As a by-product we obtain that H⁎(Simpnm(C);Q) is independent of n, thus implying rational cohomological stability. When charK>0 our methods compute He´t⁎(Simpnm;Qℓ) provided charK>n+1 and show that the étale cohomology groups in positive characteristics do not stabilize.

Commensurating actions of birational groups and groups of pseudo-automorphisms

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Property (T), is birationally conjugate to a group acting by pseudo-automorphisms on some non-empty Zariski-open subset. We apply this argument to classify groups of birational transformations of surfaces with this fixed point property up to birational conjugacy.