Ads
related to: secant variety algebra 2 formula
Search results
Results From The WOW.Com Content Network
A secant variety can be used to show the fact that a smooth projective curve can be embedded into the projective 3-space as follows. [2] Let be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if >, then there is a point p on that is not on S and so we have the projection from p to a hyperplane H, which gives the embedding :.
The twisted cubic is a projective algebraic variety. Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in ...
The union of the tangent and secant lines (the secant variety) of a twisted cubic C fill up P 3 and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the tangent and secant lines of any non-planar smooth algebraic curve is three-dimensional.
The secant variety to a projective variety is the closure of the union of all secant lines to V in . section ring The section ring or the ring of sections of a line bundle L on a scheme X is the graded ring ⊕ 0 ∞ Γ ( X , L n ) {\displaystyle \oplus _{0}^{\infty }\Gamma (X,L^{n})} .
Similarly, [3] if C is a smooth curve on the quadric surface P 1 ×P 1 with bidegree (d 1,d 2) (meaning d 1,d 2 are its intersection degrees with a fiber of each projection to P 1), since the canonical class of P 1 ×P 1 has bidegree (−2,−2), the adjunction formula shows that the canonical class of C is the intersection product of divisors ...
This connection between a geometric notion of intersection and a homological notion of a derived tensor product has been influential and led in particular to several homological conjectures in commutative algebra. Serre's Tor formula states: let X be a regular variety, V and W two subvarieties of complementary dimension such that V ∩ W is ...
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
The name "trisecant identity" refers to the geometric interpretation given by Mumford (1984, p.3.219), who used it to show that the Kummer variety of a genus g Riemann surface, given by the image of the map from the Jacobian to projective space of dimension induced by theta functions of order 2, has a 4-dimensional space of trisecants.