Twisted cubic

In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a projective variety that is not linear or a hypersurface, in fact not a complete intersection. It is the three-dimensional case of the rational normal curve, and is the of a Veronese map of degree three on the projective line. The twisted cubic is most easily given parametrically as the image of the map which assigns to the homogeneous coordinate the value In one coordinate patch of projective space, the map is simply the moment curve That is, it is the closure by a single point at infinity of the affine curve . The twisted cubic is a projective variety, defined as the intersection of three quadrics. In homogeneous coordinates on P3, the twisted cubic is the closed subscheme defined by the vanishing of the three homogeneous polynomials It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substitute x3 for X, and so on. More strongly, the homogeneous ideal of the twisted cubic C is generated by these three homogeneous polynomials of degree 2. The twisted cubic has the following properties: It is the set-theoretic complete intersection of and , but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety cannot be generated by only 2 polynomials; a minimum of 3 are needed. (An attempt to use only two polynomials make the resulting ideal not radical, since is in it, but is not). Any four points on C span P3. Given six points in P3 with no four coplanar, there is a unique twisted cubic passing through them. The union of the tangent and secant lines (the secant variety) of a twisted cubic C fill up P3 and the lines are pairwise disjoint, except at points of the curve itself.

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Glossary of algebraic geometry

This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.

Rational normal curve

In mathematics, the rational normal curve is a smooth, rational curve C of degree n in projective n-space Pn. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For n = 2 it is the plane conic Z0Z2 = Z, and for n = 3 it is the twisted cubic. The term "normal" refers to projective normality, not normal schemes. The intersection of the rational normal curve with an affine space is called the moment curve.

Degree of an algebraic variety

In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety with n hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points).

EXISTENCE OF AN UNBOUNDED NODAL HYPERSURFACE FOR SMOOTH GAUSSIAN FIELDS IN DIMENSION d >= 3

Alejandro Rivera

For the Bargmann-Fock field on R-d with d >= 3, we prove that the critical level l(c) (d) of the percolation model formed by the excursion sets {f >= l} is strictly positive. This implies that for every l sufficiently close to 0 (in particular for the noda ...

INST MATHEMATICAL STATISTICS-IMS2023

On algebraic twists with composite moduli

Philippe Michel, Yongxiao Lin

We study bounds for algebraic twists sums of automorphic coefficients by trace functions of composite moduli. ...

Dordrecht2023

ALGEBRAIC TWISTS OF GL3 x GL2 L-FUNCTIONS

Philippe Michel, Yongxiao Lin

We prove that the coefficients of a GL3 x GL2 Rankin-Selberg L-function do not correlate with a wide class of trace functions of small conductor modulo primes, generalizing the corresponding result of Fouvry, Kowalski, and Michel for GL2 and of Kowalski, L ...

Baltimore2023

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Explore les points singuliers, la loi de groupe, et l'ambiguïté dans la définition de la somme d'un point avec lui-même sur les courbes elliptiques.

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