scholarly journals Stable birational equivalence and geometric Chevalley-Warning

2013 ◽  
Vol 141 (12) ◽  
pp. 4049-4055
Author(s):  
Xia Liao
Keyword(s):  
Birational Equivalence

scholarly journals MINIMAL MODEL PROGRAM WITH SCALING AND ADJUNCTION THEORY

2013 ◽  
Vol 24 (02) ◽  
pp. 1350007 ◽  
Author(s):  
MARCO ANDREATTA
Keyword(s):  
Minimal Model ◽  
Projective Variety ◽  
Model Program ◽  
Minimal Model Program ◽  
Projective Varieties ◽  
Complex Projective Variety ◽  
Adjunction Theory ◽  
Birational Equivalence ◽  
Big Line Bundle ◽  
Complex Projective

Let (X, L) be a quasi-polarized pair, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles KX + rL for high rational numbers r. For this we run a Minimal Model Program with scaling relative to the divisor KX + rL. We give then some applications, namely the classification up to birational equivalence of quasi-polarized pairs with sectional genus 0, 1 and of embedded projective varieties X ⊂ ℙN with degree smaller than 2 codim ℙN(X) + 2.

scholarly journals CHARACTER VARIETIES OF ONCE-PUNCTURED TORUS BUNDLES WITH TUNNEL NUMBER ONE

2013 ◽  
Vol 24 (06) ◽  
pp. 1350048 ◽  
Author(s):  
KENNETH L. BAKER ◽  
KATHLEEN L. PETERSEN
Keyword(s):  
Boundary Component ◽  
Lens Space ◽  
Algebraic Sets ◽  
Character Varieties ◽  
Torus Bundles ◽  
Whitehead Link ◽  
Birational Equivalence ◽  
Punctured Torus ◽  
Tunnel Number ◽  
Trace Fields

We determine the PSL2(ℂ) and SL2(ℂ) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine "natural" models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Additionally, we study the action of the symmetries of these manifolds upon their character varieties, identify the characters of their lens space fillings, and compute the twisted Alexander polynomials for their representations to SL2(ℂ).

scholarly journals SUPERSINGULAR EDWARDS CURVES AND EDWARDS CURVE POINTS COUNTING METHOD OVER FINITE FIELD

2020 ◽  
pp. 68-88
Author(s):  
Ruslan Skuratovskii
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Algebraic Curves ◽  
Discrete Logarithm ◽  
Edwards Curves ◽  
Projective Curves ◽  
Birational Equivalence ◽  
Supersingular Curves ◽  
Edwards Curve ◽  
Embedding Degree

We consider problem of order counting of algebraic affine and projective curves of Edwards [2, 8] over the finite field $F_{p^n}$. The complexity of the discrete logarithm problem in the group of points of an elliptic curve depends on the order of this curve (ECDLP) [4, 20] depends on the order of this curve [10]. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve over a finite field. It should be noted that this method can be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. We not only find a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve $E_d [F_p]$ is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over $F_{p^n}$ in a finite field is investigated and the field characteristic, where this degree is minimal, is found. A birational isomorphism between the Montgomery curve and the Edwards curve is also constructed. A one-to-one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over $F_{p^n}$.

scholarly journals Birational equivalence of reduced graphs

1998 ◽  
Vol 124 (1-3) ◽  
pp. 173-199 ◽  
Author(s):  
Carlos Marijuán
Keyword(s):  
Birational Equivalence

On the Birational Equivalence of Curves Under Specialization

1957 ◽  
Vol 79 (3) ◽  
pp. 649 ◽  
Author(s):  
Wei-Liang Chow ◽  
Serge Lang
Keyword(s):  
Birational Equivalence

scholarly journals On the Grothendieck ring of varieties

2015 ◽  
Vol 158 (3) ◽  
pp. 477-486
Author(s):  
AMIT KUBER
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Equivalence Classes ◽  
Closed Field ◽  
Associated Graded Ring ◽  
Grothendieck Ring ◽  
Algebraically Closed Field ◽  
Monoid Ring ◽  
Birational Equivalence ◽  
Multiplicative Monoid

AbstractLet K0(Vark) denote the Grothendieck ring of k-varieties over an algebraically closed field k. Larsen and Lunts asked if two k-varieties having the same class in K0(Vark) are piecewise isomorphic. Gromov asked if a birational self-map of a k-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb{Z}$[$\mathfrak{B}$] where $\mathfrak{B}$ denotes the multiplicative monoid of birational equivalence classes of irreducible k-varieties.

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