Monodromy of projections of hypersurfaces

Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group $$S_d$$ S d . We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least 2, except possibly for a special class of hypersurfaces with singular locus linear in codimension 1. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points.

A Polynomial Sieve and Sums of Deligne Type

Let $f\in \mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in \mathbb{Z}[X_{0},...,X_{n}]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we present a new bound for $N(f,F,B):=|\{\textbf{x}\in \mathbb{Z}^{n+1}:\max _{0\leq i\leq n}|x_{i}|\leq B,\exists t\in \mathbb{Z}\textrm{ such that}\ f(t)=F(\textbf{x})\}|.$ To do this, we introduce a generalization of the power sieve [14, 28] and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables.

Counting Multiplicities in a Hypersurface over Number Fields

We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.

Thom Polynomials and the Green–Griffiths–Lang Conjecture for Hypersurfaces with Polynomial Degree

Green and Griffiths [25] and Lang [29] conjectured that for every complex projective algebraic variety X of general type there exists a proper algebraic subvariety of X containing all nonconstant entire holomorphic curves $f:{\mathbb{C}} \to X$. We construct a compactification of the invariant jet differentials bundle over complex manifolds motivated by an algebraic model of Morin singularities and we develop an iterated residue formula using equivariant localisation for tautological integrals over it. Using this we show that the polynomial Green–Griffiths–Lang conjecture for a generic projective hypersurface of degree $\deg (X)>2n^{9}$ follows from a positivity conjecture for Thom polynomials of Morin singularities.

A geometric approach to Orlov’s theorem

A famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations. In this article, I implement a proposal of E. Segal to prove Orlov’s theorem in the Calabi–Yau setting using a globalization of the category of graded matrix factorizations (graded D-branes). Let X⊂ℙ be a projective hypersurface. Segal has already established an equivalence between Orlov’s category of graded matrix factorizations and the category of graded D-branes on the canonical bundle Kℙ to ℙ. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on Kℙ and Dbcoh(X). This can be achieved directly, as well as by deforming Kℙ to the normal bundle of X⊂Kℙ and invoking a global version of Knörrer periodicity. We also discuss an equivalence between graded D-branes on a general smooth quasiprojective variety and on the formal neighborhood of the singular locus of the zero fiber of the potential.

Polar cremona transformations and monodromy of polynomials

Consider the gradient map associated to any non-constant homogeneous polynomial f ∈ ℂ[ x0 , ..., xn ] of degree d , defined by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\varphi _f = grad (f):D(f) \to \mathbb{P}^n ,(x_0 : \ldots :x_n ) \to (f_0 (x): \ldots :f_n (x))$$ \end{document} where D ( f ) = { x ∈ ℙ n ; f ( x ) ≠ 0} is the principal open set associated to f and fi = ∂f / ∂xi . This map corresponds to polar Cremona transformations. In Proposition 3.4 we give a new lower bound for the degree d ( f ) of ϕ f under the assumption that the projective hypersurface V : f = 0 has only isolated singularities. When d ( f ) = 1, Theorem 4.2 yields very strong conditions on the singularities of V .

PROJECTIVE SPECTRUM IN BANACH ALGEBRAS

For a tuple A = (A1, A2, …, An) of elements in a unital algebra [Formula: see text] over ℂ, its projective spectrumP(A) or p(A) is the collection of z ∈ ℂn, or respectively z ∈ ℙn-1 such that A(z) = z1A1 + z2A2 + ⋯ + znAn is not invertible in [Formula: see text]. In finite dimensional case, projective spectrum is a projective hypersurface. When A is commuting, P(A) looks like a bundle over the Taylor spectrum of A. In the case [Formula: see text] is reflexive or is a C*-algebra, the projective resolvent setPc(A) := ℂn \ P(A) is shown to be a disjoint union of domains of holomorphy. [Formula: see text]-valued 1-form A-1(z)dA(z) reveals the topology of Pc(A), and a Chern–Weil type homomorphism from invariant multilinear functionals to the de Rham cohomology [Formula: see text] is established.

Differential equations on complex projective hypersurfaces of low dimension

Let n=2,3,4,5 and let X be a smooth complex projective hypersurface of $\mathbb {P}^{n+1}$. In this paper we find an effective lower bound for the degree of X, such that every holomorphic entire curve in X must satisfy an algebraic differential equation of order k=n=dim X, and also similar bounds for order k>n. Moreover, for every integer n≥2, we show that there are no such algebraic differential equations of order k<n for a smooth hypersurface in $\mathbb {P}^{n+1}$.