scholarly journals Monodromy of projections of hypersurfaces

2021 ◽  
Author(s):  
Maria Gioia Cifani ◽  
Alice Cuzzucoli ◽  
Riccardo Moschetti
Keyword(s):  
Symmetric Group ◽  
Special Class ◽  
Monodromy Group ◽  
Singular Locus ◽  
Finite Union ◽  
Smooth Variety ◽  
Linear Spaces ◽  
Projective Hypersurface ◽  
General Projection ◽  
Complex Projective

AbstractLet 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.

On complex projective hypersurfaces

1981 ◽  
Vol 24 (2) ◽  
pp. 91-97
Author(s):  
J. W. Bruce
Keyword(s):  
Financial Support ◽  
Monodromy Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Isolated Singularity ◽  
Projective Hypersurface ◽  
Veronese Variety ◽  
Monodromy Groups ◽  
Necessary And Sufficient ◽  
Complex Projective

In this paper we prove various results concerning monodromy groups associated with nonsingular complex projective hypersurfaces. Most of these results are already known but proofs are either unavailable or are algebraic and require a lot of machinery. The groups in question are those obtained from the second Lefschetz theorem (see (1)) applied to (a) the general Veronese variety, (b) a nonsingular projective hypersurface. By embedding the monodromy group of an extraordinary local isolated singularity (discovered by Libgober (8)) in these global monodromy groups we obtain necessary and sufficient conditions for the global groups to be finite. For case (a) we also obtain information on the structure of the dual to the Veronese variety which is of use when considering the monodromy group. The author gratefully acknowledges the financial support of the Stiftung Volkswagenwerk for a vist to the IHES during which this paper was written.

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

2017 ◽  
Vol 2019 (22) ◽  
pp. 7037-7092
Author(s):  
Gergely Bérczi
Keyword(s):  
Algebraic Variety ◽  
General Type ◽  
Algebraic Model ◽  
Holomorphic Curves ◽  
Complex Manifolds ◽  
Projective Hypersurface ◽  
Polynomial Degree ◽  
Residue Formula ◽  
Thom Polynomials ◽  
Complex Projective

Abstract 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.

scholarly journals Construction of the Fuchs equation with four given finite critical points and a given reducible monodromy group in the resonance case

2019 ◽  
Vol 55 (2) ◽  
pp. 199-206
Author(s):  
V. V. Amel’kin ◽  
M. N. Vasilevich
Keyword(s):  
Critical Points ◽  
Monodromy Group ◽  
Projective Line ◽  
Resonance Case ◽  
Nonsingular Transformation ◽  
Triangular Form ◽  
Rank 2 ◽  
Linear Differential ◽  
Complex Projective ◽  
Monodromy Matrices

One inverse problem of the analytic theory of linear differential equations is considered. Namely, the completely integrable Fuchs equation with four given finite critical points and a given reducible monodromy group of rank 2 on the complex projective line is constructed. Reducibility of the monodromy group of rank 2 means that 2×2-monodromy matrices (the generators of the monodromy group) can be simultaneously reduced by a linear nonsingular transformation to an upper triangular form. In so doing we study the case when the eigenvalue ξj of the diagonal matrix of the monodromy formal exponent at a corresponding Fuchs critical point is equal to an integer different from zero (resonance takes place).

scholarly journals A geometric approach to Orlov’s theorem

2012 ◽  
Vol 148 (5) ◽  
pp. 1365-1389 ◽  
Author(s):  
Ian Shipman
Keyword(s):  
Normal Bundle ◽  
Geometric Approach ◽  
Singular Locus ◽  
Matrix Factorizations ◽  
Canonical Bundle ◽  
Algebraic Construction ◽  
Famous Theorem ◽  
Projective Hypersurface ◽  
Coherent Sheaves ◽  
Formal Neighborhood

AbstractA 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.

scholarly journals A new class of fractional type set-valued functions

2019 ◽  
Vol 35 (1) ◽  
pp. 79-84
Author(s):  
ALEXANDRU ORZAN ◽  
Keyword(s):  
Special Class ◽  
Convex Sets ◽  
Linear Spaces ◽  
New Class ◽  
Finite Dimensional ◽  
Affine Functions ◽  
Real Linear ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Fractional Type

The so-called ratios of affine functions, introduced by Rothblum (1985) in the framework of finite-dimensional Euclidean spaces, represent a special class of fractional type vector-valued functions, which transform convex sets into convex sets. The aim of this paper is to show that a similar convexity preserving property holds within a new class of fractional type set-valued functions, acting between any real linear spaces.

scholarly journals Connected Components of the Hurwitz Space for the Symmetric Group of Degree 7

2020 ◽  
Vol 24 (2) ◽  
pp. 129
Author(s):  
Haval M. Mohammed Salih
Keyword(s):  
Symmetric Group ◽  
Monodromy Group ◽  
Riemann Sphere ◽  
Complete Classification ◽  
Connected Components ◽  
Branch Points ◽  
Genus Zero ◽  
Computational Tools ◽  
Hurwitz Space

The Hurwitz space  is the space of genus  covers of the Riemann sphere  with branch points and the monodromy group . Let be the symmetric group . In this paper, we enumerate the connected components of . Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of . This work gives us the complete classification of  primitive genus zero symmetric group of degree seven. 

scholarly journals Counting Multiplicities in a Hypersurface over Number Fields

2018 ◽  
Vol 25 (03) ◽  
pp. 437-458
Author(s):  
Hao Wen ◽  
Chunhui Liu
Keyword(s):  
Number Field ◽  
Upper Bound ◽  
Counting Function ◽  
Singular Locus ◽  
Number Fields ◽  
Upper Bounds ◽  
Fixed Degree ◽  
Projective Hypersurface ◽  
Field Of Definition

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.

scholarly journals Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties

1981 ◽  
Vol 83 ◽  
pp. 213-233 ◽  
Author(s):  
Junjiro Noguchi
Keyword(s):  
Meromorphic Functions ◽  
Holomorphic Mapping ◽  
Holomorphic Curves ◽  
Compact Complex Manifold ◽  
Algebraic Varieties ◽  
Second Main Theorem ◽  
Smooth Variety ◽  
Complex Projective ◽  
Logarithmic Derivatives ◽  
The Second Main Theorem

Nevanlinna’s lemma on logarithmic derivatives played an essential role in the proof of the second main theorem for meromorphic functions on the complex plane C (cf., e.g., [17]). In [19, Lemma 2.3] it was generalized for entire holomorphic curves f: C → M in a compact complex manifold M (Lemma 2.3 in [19] is still valid for non-Kähler M). Here we call, in general, a holomorphic mapping from a domain of C or a Riemann surface into M a holomorphic curve in M, and sometimes use it in the sense of its image if no confusion occurs. Applying the above generalized lemma on logarithmic derivatives to holomorphic curves f: C → V in a complex projective algebraic smooth variety V and making use of Ochiai [22, Theorem A], we had an inequality of the second main theorem type for f and divisors on V (see [19, Main Theorem] and [20]). Other generalizations of Nevanlinna’s lemma on logarithmic derivatives were obtained by Nevanlinna [16], Griffiths-King [10, § 9] and Vitter [23].

scholarly journals Differential equations on complex projective hypersurfaces of low dimension

2008 ◽  
Vol 144 (4) ◽  
pp. 920-932 ◽  
Author(s):  
Simone Diverio
Keyword(s):  
Differential Equation ◽  
Lower Bound ◽  
Differential Equations ◽  
Algebraic Differential Equation ◽  
Projective Hypersurface ◽  
Algebraic Differential Equations ◽  
Smooth Hypersurface ◽  
Smooth Complex ◽  
Low Dimension ◽  
Complex Projective

AbstractLet 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 $\mathbb {P}^{n+1}$.

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