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Determining the Topology of Real Algebraic Surfaces
... An implicit real algebraic surface (or curve, or hypersurface) S in Ru with degree d is defined by f (x1 , x2 , · · · , xu ) = 0 where f (x1 , x2 , · · · , xu ) ∈ Q[x1 , x2 , · · ·, xu ] is a polynomial of degree d, and R and Q are the fields of real and rational numbers, respectively. Determining the ...
... An implicit real algebraic surface (or curve, or hypersurface) S in Ru with degree d is defined by f (x1 , x2 , · · · , xu ) = 0 where f (x1 , x2 , · · · , xu ) ∈ Q[x1 , x2 , · · ·, xu ] is a polynomial of degree d, and R and Q are the fields of real and rational numbers, respectively. Determining the ...
6. Divisors Definition 6.1. We say that a scheme X is regular in
... This implies that if P and P 0 are two smooth points of C then P and P 0 are not linearly equivalent. It follows, with a little bit of work, that all linear equivalences on C are generated by the linear equivalences above. The normalisation of C is isomorphic to P1 ; on the affine piece where Z 6= 0 ...
... This implies that if P and P 0 are two smooth points of C then P and P 0 are not linearly equivalent. It follows, with a little bit of work, that all linear equivalences on C are generated by the linear equivalences above. The normalisation of C is isomorphic to P1 ; on the affine piece where Z 6= 0 ...
VARIATIONS ON A QUESTION OF LARSEN AND LUNTS 1
... a positive answer too. Furthermore, we give partial (i.e., in dimensions 1 or 2), but unconditional, solutions to each of these two problems (see Propositions 4.2 and 4.6). In §3, we complete, for k-varieties of dimension one or two and in the non-“smooth and projective” context (see Theorem 3.3), t ...
... a positive answer too. Furthermore, we give partial (i.e., in dimensions 1 or 2), but unconditional, solutions to each of these two problems (see Propositions 4.2 and 4.6). In §3, we complete, for k-varieties of dimension one or two and in the non-“smooth and projective” context (see Theorem 3.3), t ...
7. Divisors Definition 7.1. We say that a scheme X is regular in
... This implies that if P and P 0 are two smooth points of C then P and P 0 are not linearly equivalent. It follows, with a little bit of work, that all linear equivalences on C are generated by the linear equivalences above. The normalisation of C is isomorphic to P1 ; on the affine piece where Z 6= 0 ...
... This implies that if P and P 0 are two smooth points of C then P and P 0 are not linearly equivalent. It follows, with a little bit of work, that all linear equivalences on C are generated by the linear equivalences above. The normalisation of C is isomorphic to P1 ; on the affine piece where Z 6= 0 ...
N-Symmetry Direction Fields on Surfaces of Arbitrary Genus
... structure is computed from the Morse complex of a smooth harmonic function, with user-controllable number and configuration of singularities. The gradient of the harmonic function is a direction field (with the same singular points as the harmonic functions). It was used in [Zelinka and Garland 2004 ...
... structure is computed from the Morse complex of a smooth harmonic function, with user-controllable number and configuration of singularities. The gradient of the harmonic function is a direction field (with the same singular points as the harmonic functions). It was used in [Zelinka and Garland 2004 ...
Derived splinters in positive characteristic
... class of singularities is closely related to other classes of singularities, the so-called F-singularities, defined using the Frobenius action. For example, locally excellent affine Q-Gorenstein splinters are F-regular by [Sin99], which builds on the Gorenstein case of [HH94]; see also Example 2.4 b ...
... class of singularities is closely related to other classes of singularities, the so-called F-singularities, defined using the Frobenius action. For example, locally excellent affine Q-Gorenstein splinters are F-regular by [Sin99], which builds on the Gorenstein case of [HH94]; see also Example 2.4 b ...
MARCH 10 Contents 1. Strongly rational cones 1 2. Normal toric
... Let σ be a strongly convex cone and let ρ be an edge. Then ρ = R+ e, where e is the generator of the semigroup ρ ∩ N. The lattice point e is called the ray of ρ. Any strongly convex cone is generated by the ray generating its edges, called a minimal set of generators. Moreover |{ edges of σ̌}| ≥ n. ...
... Let σ be a strongly convex cone and let ρ be an edge. Then ρ = R+ e, where e is the generator of the semigroup ρ ∩ N. The lattice point e is called the ray of ρ. Any strongly convex cone is generated by the ray generating its edges, called a minimal set of generators. Moreover |{ edges of σ̌}| ≥ n. ...
RULED SURFACES WITH NON-TRIVIAL SURJECTIVE
... fibration h : X → C onto a non-singular curve C. The fibers of π dominate C. Hence C P1 . Let D be a general fiber of h. Then D2 = 0 and π(D) = B. (2) =⇒ (3). If there is a section C0 of π with C02 < 0, then any other irreducible curve C with π(C) = B is linearly equivalent to aC0 + π ∗E for some a > ...
... fibration h : X → C onto a non-singular curve C. The fibers of π dominate C. Hence C P1 . Let D be a general fiber of h. Then D2 = 0 and π(D) = B. (2) =⇒ (3). If there is a section C0 of π with C02 < 0, then any other irreducible curve C with π(C) = B is linearly equivalent to aC0 + π ∗E for some a > ...
The number of conjugacy classes of elements of the Cremona group
... We are now able to prove Theorem 1.2, i.e. to show the existence of infinitely many conjugacy classes of elements of order n in the Cremona group, for any even integer n and for n = 3, 5. Proof of Theorem 1.2. — First of all, taking some non-rational curve Γ, any birational transformation sends Γ on ...
... We are now able to prove Theorem 1.2, i.e. to show the existence of infinitely many conjugacy classes of elements of order n in the Cremona group, for any even integer n and for n = 3, 5. Proof of Theorem 1.2. — First of all, taking some non-rational curve Γ, any birational transformation sends Γ on ...
Cellular Resolutions of Monomial Modules
... two comparable faces F 0 ⊂ F of the complex X have distinct degrees aF 6= aF 0 . The simplest example of a cellular resolution is the Taylor resolution for monomial ideals [Tay]. The Taylor resolution is easily generalized to arbitrary monomial modules M as follows. Let { mj | j ∈ I } be the minimal ...
... two comparable faces F 0 ⊂ F of the complex X have distinct degrees aF 6= aF 0 . The simplest example of a cellular resolution is the Taylor resolution for monomial ideals [Tay]. The Taylor resolution is easily generalized to arbitrary monomial modules M as follows. Let { mj | j ∈ I } be the minimal ...
Why Use Curves? - cloudfront.net
... Curve Techniques: Center-Line Symmetry When using the Revolve tool, having two CVs perpendicular to the rotation axis gives a smooth transition. ...
... Curve Techniques: Center-Line Symmetry When using the Revolve tool, having two CVs perpendicular to the rotation axis gives a smooth transition. ...
Toric Varieties
... a prime binomial ideal in a polynomial ring. More generally, a toric variety can be described by a multigraded ring together with an irrelevant ideal. The importance of toric varieties comes from this dictionary between algebraic spaces, discrete geometric objects such as cones and polytopes, and mu ...
... a prime binomial ideal in a polynomial ring. More generally, a toric variety can be described by a multigraded ring together with an irrelevant ideal. The importance of toric varieties comes from this dictionary between algebraic spaces, discrete geometric objects such as cones and polytopes, and mu ...
PM 464
... show that this is not an algebraic set. 2. Properties (3), (4), and (5) show that the collection of algebraic sets form the closed sets of a topology on An . This topology is known as the Zariski topology 3. The Zariski topology is strictly weaker than the metric topology (for | = Q, R, C), i.e. Zar ...
... show that this is not an algebraic set. 2. Properties (3), (4), and (5) show that the collection of algebraic sets form the closed sets of a topology on An . This topology is known as the Zariski topology 3. The Zariski topology is strictly weaker than the metric topology (for | = Q, R, C), i.e. Zar ...
WHICH ARE THE SIMPLEST ALGEBRAIC VARIETIES? Contents 1
... Figure 3. Tangent method for 4y 2 = x4 − 5x2 + 5 in 4 points; thus the above tangent method does not work. One can, however, modify it as follows. It is easy to see that every parabola of the form y = x2 +Ax+B intersects H4 in at most 3 points, and for every (x0 , y0 ) ∈ H4 there is a unique parabol ...
... Figure 3. Tangent method for 4y 2 = x4 − 5x2 + 5 in 4 points; thus the above tangent method does not work. One can, however, modify it as follows. It is easy to see that every parabola of the form y = x2 +Ax+B intersects H4 in at most 3 points, and for every (x0 , y0 ) ∈ H4 there is a unique parabol ...
Computing self-intersection curves of rational ruled surfaces
... A singular point of a surface is a point on the surface where the tangent plane of the surface is not uniquely defined. The singular locus of a surface is the set of all the singular points on the surface. In general, the singular locus of a surface consists of a finite number of isolated points on th ...
... A singular point of a surface is a point on the surface where the tangent plane of the surface is not uniquely defined. The singular locus of a surface is the set of all the singular points on the surface. In general, the singular locus of a surface consists of a finite number of isolated points on th ...
4. Topic
... The intersect of a plane through 0 with the sphere is a great circle, which defines a line in RP2 in the following manner. A plane through 0 in R3 is the set ...
... The intersect of a plane through 0 with the sphere is a great circle, which defines a line in RP2 in the following manner. A plane through 0 in R3 is the set ...
SOME GEOMETRIC PROPERTIES OF CLOSED SPACE CURVES
... an (n + 1)-gon with equal sides and lying in a hyperplane. In particular, a rhombus can be inscribed in each circle smoothly embedded in R3 . Remarks. 1. In the paper [2], Shnirelman gave two proofs of the fact that a square can be inscribed in every smooth Jordan curve in the plane. The second pro ...
... an (n + 1)-gon with equal sides and lying in a hyperplane. In particular, a rhombus can be inscribed in each circle smoothly embedded in R3 . Remarks. 1. In the paper [2], Shnirelman gave two proofs of the fact that a square can be inscribed in every smooth Jordan curve in the plane. The second pro ...
Intersection homology
... 1970s; the goal was to produce a homology theory that behaves as well for singular spaces as it does for manifolds, in the sense that basic properties such as Poincaré duality and the Lefschetz theorems hold even for singular projective varieties. Intersection homology is defined for a class of spa ...
... 1970s; the goal was to produce a homology theory that behaves as well for singular spaces as it does for manifolds, in the sense that basic properties such as Poincaré duality and the Lefschetz theorems hold even for singular projective varieties. Intersection homology is defined for a class of spa ...
lecture 2
... functions of a variable or parameter such as t. A curve in the plane has the form C(t) = (x(t), y(t)), and a curve in space has the form C(t) = (x(t), y(t), z(t)). The functions x(t), y(t), and z(t) are called the coordinate functions. The image of C(t) is called the trace of C, and C(t) is called a ...
... functions of a variable or parameter such as t. A curve in the plane has the form C(t) = (x(t), y(t)), and a curve in space has the form C(t) = (x(t), y(t), z(t)). The functions x(t), y(t), and z(t) are called the coordinate functions. The image of C(t) is called the trace of C, and C(t) is called a ...
Chapter 5
... • invariant under affine as well as perspective transformations (affine invariance and projective invariance). • single mathematical form for both analytical and free-form shapes. • flexibility to design variety of shapes. • less memory is used when storing shapes in comparison to other methods. • n ...
... • invariant under affine as well as perspective transformations (affine invariance and projective invariance). • single mathematical form for both analytical and free-form shapes. • flexibility to design variety of shapes. • less memory is used when storing shapes in comparison to other methods. • n ...
Chapter 7 - U.I.U.C. Math
... conclude that (2) holds. Since (3) is a special case of (2), we have (2) implies (3). If (4) holds, construct a projective resolution of M in the usual way, but pause at Xn−1 and terminate the sequence with 0 → Kn−1 → Xn−1 . By hypothesis, Kn−1 is projective, and this gives (4) implies (1). The main ...
... conclude that (2) holds. Since (3) is a special case of (2), we have (2) implies (3). If (4) holds, construct a projective resolution of M in the usual way, but pause at Xn−1 and terminate the sequence with 0 → Kn−1 → Xn−1 . By hypothesis, Kn−1 is projective, and this gives (4) implies (1). The main ...
the isoperimetric problem on some singular surfaces
... best. For double discs and cylinders, we use spherical Schwarz symmetrization to limit complexity and then show that the minimizers have one component by creating illegal singularities in multi-component competitors. For general dimension double discs, we use Schwarz spherical symmetrization combine ...
... best. For double discs and cylinders, we use spherical Schwarz symmetrization to limit complexity and then show that the minimizers have one component by creating illegal singularities in multi-component competitors. For general dimension double discs, we use Schwarz spherical symmetrization combine ...
THE MOVING CURVE IDEAL AND THE REES
... In other words, the syzygy module Syz(a, b, c) is free with generators p, q of degree µ and n − µ. We assume µ ≤ n − µ, and we can regard p, q as moving lines p = Ax + By + Cz q = Ex + F y + Gz. On easily sees that p, q are the degree 1 generators of the moving curve ideal MC. Question What can we s ...
... In other words, the syzygy module Syz(a, b, c) is free with generators p, q of degree µ and n − µ. We assume µ ≤ n − µ, and we can regard p, q as moving lines p = Ax + By + Cz q = Ex + F y + Gz. On easily sees that p, q are the degree 1 generators of the moving curve ideal MC. Question What can we s ...
ON THE IRREDUCIBILITY OF SECANT CONES, AND
... W is normal, then so is a general fibre of h: this can be seen, e.g. using Serre’s criterion, or alternatively, use [G], Thm 12.2.4, which says, in the scheme-theoretic context, that the set N (h) of points t ∈ T such that h−1 (t) is normal is open; when W is normal, N (h) contains the generic point ...
... W is normal, then so is a general fibre of h: this can be seen, e.g. using Serre’s criterion, or alternatively, use [G], Thm 12.2.4, which says, in the scheme-theoretic context, that the set N (h) of points t ∈ T such that h−1 (t) is normal is open; when W is normal, N (h) contains the generic point ...