scholarly journals Constructions of Brauer-Severi Varieties and Norm Hypersurfaces

1990 ◽  
Vol 42 (2) ◽  
pp. 230-238 ◽  
Author(s):  
Ming-Chang Kang
Keyword(s):  
Function Field ◽  
Splitting Field ◽  
Severi Varieties ◽  
Severi Variety ◽  
Central Simple

Let k be any field, A a central simple k-algebra of degree m (i.e., dimk A = m2). Several methods of constructing the generic splitting fields for A are proposed and Saltman proves that these methods result in almost the same generic splitting field [8, Theorems 4.2 and 4.4]. In fact, the generic splitting field constructed by Roquette [7] is the function field of the Brauer- Severi variety Vm(A) while the generic splitting field constructed by Heuser and Saltman [4 and 8] is the function field of the norm surface W(A). In this paper, to avoid possible confusion about the dimension, we shall call it the norm hypersurface instead of the norm surface.

scholarly journals Local–global principle for reduced norms over function fields of -adic curves

2017 ◽  
Vol 154 (2) ◽  
pp. 410-458 ◽  
Author(s):  
R. Parimala ◽  
R. Preeti ◽  
V. Suresh
Keyword(s):  
Local Field ◽  
Function Field ◽  
Function Fields ◽  
Residue Field ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Hasse Principle ◽  
Central Simple ◽  
Global Principle ◽  
Reduced Norm

Let $K$ be a (non-archimedean) local field and let $F$ be the function field of a curve over $K$. Let $D$ be a central simple algebra over $F$ of period $n$ and $\unicode[STIX]{x1D706}\in F^{\ast }$. We show that if $n$ is coprime to the characteristic of the residue field of $K$ and $D\cdot (\unicode[STIX]{x1D706})=0$ in $H^{3}(F,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$, then $\unicode[STIX]{x1D706}$ is a reduced norm from $D$. This leads to a Hasse principle for the group $\operatorname{SL}_{1}(D)$, namely, an element $\unicode[STIX]{x1D706}\in F^{\ast }$ is a reduced norm from $D$ if and only if it is a reduced norm locally at all discrete valuations of $F$.

scholarly journals SEVERI VARIETIES AND BRANCH CURVES OF ABELIAN SURFACES OF TYPE (1, 3)

2002 ◽  
Vol 13 (03) ◽  
pp. 227-244 ◽  
Author(s):  
H. LANGE ◽  
E. SERNESI
Keyword(s):  
Linear System ◽  
Plane Curve ◽  
Main Idea ◽  
Abelian Surface ◽  
Severi Varieties ◽  
Severi Variety ◽  
System L ◽  
Discriminant Curve ◽  
Branch Curve

A polarized abelian surface (A, L) of type (1, 3) induces a 6:1 covering of A onto the projective plane with branch curve, a plane curve B of degree 18. The main result of the paper is that for a general abelian surface of type (1, 3), the curve B is irreducible and reduced and admits 72 cusps, 36 nodes or tacnodes, each tacnode counting as 2 nodes, 72 flexes and 36 bitangents. The main idea of the proof is that for a general (A, L) the discriminant curve in the linear system |L| coincides with the closure of the Severi variety of curves in |L| admitting a node and is dual to the curve B in the sense of projective geometry.

Non-hyperbolic splitting of quadratic pairs

2015 ◽  
Vol 14 (10) ◽  
pp. 1550138 ◽  
Author(s):  
Karim Johannes Becher ◽  
Andrew Dolphin
Keyword(s):  
Quaternion Algebra ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Splitting Field ◽  
Base Field ◽  
Central Simple

We show that a non-hyperbolic quadratic pair on a central simple algebra Brauer equivalent to a quaternion algebra stays non-hyperbolic over some splitting field of the quaternion algebra. This extends a result previously only known for fields of characteristic different from two. Our presentation is free from restrictions on the characteristic of the base field.

scholarly journals Diagonal quartic surfaces and transcendental elements of the Brauer group

2010 ◽  
Vol 9 (4) ◽  
pp. 769-798 ◽  
Author(s):  
Evis Ieronymou
Keyword(s):  
Function Field ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Weak Approximation ◽  
Brauer Group ◽  
Quartic Surface ◽  
Simple Algebras ◽  
Central Simple ◽  
Quartic Surfaces ◽  
Torsion Part

ABSTRACTWe exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transcendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

Glider Brauer-Severi varieties of central simple algebras

2020 ◽  
Vol 553 ◽  
pp. 175-210
Author(s):  
Frederik Caenepeel ◽  
Fred Van Oystaeyen
Keyword(s):  
Central Simple Algebras ◽  
Severi Varieties ◽  
Simple Algebras ◽  
Central Simple

scholarly journals A Characterization of Secant Varieties of Severi Varieties Among Cubic Hypersurfaces

2020 ◽  
Author(s):  
Baohua Fu ◽  
Yewon Jeong ◽  
Fyodor L Zak
Keyword(s):  
Lie Algebra ◽  
Singular Locus ◽  
Secant Varieties ◽  
Secant Variety ◽  
Severi Varieties ◽  
Severi Variety ◽  
Cubic Hypersurfaces

Abstract It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.

scholarly journals On the section conjecture and Brauer–Severi varieties

2021 ◽  
Author(s):  
Giulio Bresciani
Keyword(s):  
Number Field ◽  
Rational Map ◽  
Severi Varieties ◽  
Severi Variety ◽  
Weil Restriction ◽  
Positive Genus

AbstractJ. Stix proved that a curve of positive genus over $$\mathbb {Q}$$ Q which maps to a non-trivial Brauer–Severi variety satisfies the section conjecture. We prove that, if X is a curve of positive genus over a number field k and the Weil restriction $$R_{k/\mathbb {Q}}X$$ R k / Q X admits a rational map to a non-trivial Brauer–Severi variety, then X satisfies the section conjecture. As a consequence, if X maps to a Brauer–Severi variety P such that the corestriction $${\text {cor}}_{k/\mathbb {Q}}([P])\in {\text {Br}}(\mathbb {Q})$$ cor k / Q ( [ P ] ) ∈ Br ( Q ) is non-trivial, then X satisfies the section conjecture.

scholarly journals Relative Node Polynomials for Plane Curves

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Florian Block
Keyword(s):  
Recent Work ◽  
Plane Curves ◽  
Explicit Formulas ◽  
Severi Varieties ◽  
Severi Variety ◽  
Nous Montrons ◽  
International Audience

International audience We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined ``relative node polynomial'' in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary δ , and use it to present explicit formulas for δ ≤ 6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ . Nous généralisons les travaux récents de Fomin et Mikhalkin sur des formules polynomiales pour les degrés de Severi. Le degré de la variété de Severi des courbes planes de degré d et à δ nœuds est donné par un polynôme en d , pour δ fixé et d assez grand. Nous étendons ce résultat aux variétés de Severi généralisées paramétrant les courbes planes et qui, en outre, satisfont à des conditions de tangence d'ordres donnés avec une droite fixée. Nous montrons que les degrés de ces variétés, rééchelonnés de manière appropriée, sont donnés par un ``polynôme de noeud relatif'', défini combinatoirement, en les ordres de tangence, dès que ceux-ci sont assez grands. Nous décrivons une méthode pour calculer ces polynômes pour delta arbitraire, et l'utilisons pour présenter des formules explicites pour δ ≤ 6 . Nous donnons aussi un seuil pour la polynomialité, et calculons les premiers termes dominants pour tout δ .

scholarly journals Universal Polynomials for Severi Degrees of Toric Surfaces

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Florian Block
Keyword(s):  
Tropical Geometry ◽  
Large Family ◽  
Plane Curves ◽  
Toric Surfaces ◽  
Combinatorial Objects ◽  
Severi Varieties ◽  
Severi Variety ◽  
Nous Montrons ◽  
International Audience ◽  
Floor Diagrams

International audience The Severi variety parametrizes plane curves of degree $d$ with $\delta$ nodes. Its degree is called the Severi degree. For large enough $d$, the Severi degrees coincide with the Gromov-Witten invariants of $\mathbb{CP}^2$. Fomin and Mikhalkin (2009) proved the 1995 conjecture that for fixed $\delta$, Severi degrees are eventually polynomial in $d$. In this paper, we study the Severi varieties corresponding to a large family of toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. More surprisingly, we show that the Severi degrees are also eventually polynomial "as a function of the surface". Our strategy is to use tropical geometry to express Severi degrees in terms of Brugallé and Mikhalkin's floor diagrams, and study those combinatorial objects in detail. An important ingredient in the proof is the polynomiality of the discrete volume of a variable facet-unimodular polytope. La variété de Severi paramétrise les courbes planes de degré $d$ avec $\delta$ nœuds. Son degré s'appelle le degré de Severi. Pour $d$ assez grand, les degrés de Severi coïncident avec les invariants de Gromov-Witten de $\mathbb{CP}^2$. Fomin et Mikhalkin (2009) ont prouvé une conjecture de 1995 que pour $\delta$ fixé, les degrés de Severi sont à terme des polynômes en $d$. Nous étudions les variétés de Severi correspondant à une large famille de surfaces toriques. Nous prouvons le résultat analogue que les degrés de Severi sont à terme des fonctions polynomiales du multidegré. De manière plus surprenante, nous montrons que les degrés de Severi sont à terme des polynômes en tant que "fonction de la surface''. Notre stratégie est d'utiliser la géométrie tropicale pour exprimer les degrés de Severi en fonction des "floor diagrams" de Brugallé et Mikhalkin, et d'utiliser ces objets combinatoires en détail. Un autre ingrédient important de la preuve est la polynomialité du volume discret d'un polytope face-unimodulaire variable.

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