Abelian Varieties

... connected projective2 variety with a group structure defined by regular maps. Definition (d) does generalize, but with a caution. If A is an abelian variety over C, then A.C/ Cg = for some lattice in Cg (isomorphism simultaneously of complex manifolds and of groups). However, when g > 1, the qu ...

... connected projective2 variety with a group structure defined by regular maps. Definition (d) does generalize, but with a caution. If A is an abelian variety over C, then A.C/ Cg = for some lattice in Cg (isomorphism simultaneously of complex manifolds and of groups). However, when g > 1, the qu ...

TopoCheck - Sinergise

... A loop back or self-intersecting polygon is when: 1. The boundary of the polygon crosses itself. This error is sometimes described as a ‘Butterfly’ or ‘Figure of Eight’ polygon, or 2. The line recrosses a vertex in a different direction. These two events are illustrated in the diagrams to the left ...

... A loop back or self-intersecting polygon is when: 1. The boundary of the polygon crosses itself. This error is sometimes described as a ‘Butterfly’ or ‘Figure of Eight’ polygon, or 2. The line recrosses a vertex in a different direction. These two events are illustrated in the diagrams to the left ...

Minimal surfaces from circle patterns: Geometry from

... embeddedness. Kapouleas [21] proved the existence of new embedded examples using a new method. He considered ﬁnitely many catenoids with the same axis and planes orthogonal to this axis and showed that one can desingularize the circles of intersection by deformed Scherk towers. This existence result ...

... embeddedness. Kapouleas [21] proved the existence of new embedded examples using a new method. He considered ﬁnitely many catenoids with the same axis and planes orthogonal to this axis and showed that one can desingularize the circles of intersection by deformed Scherk towers. This existence result ...

Introduction to Conjugate Plateau Constructions

... where conjugate minimal pieces have been obtained by solving degree arguments, for example in [KPS], [Po], a trivial perturbation argument shows that constant mean curvature deformations of the constructed minimal surfaces also exist. Constant mean curvature one surfaces in H3 , also called Bryant ...

... where conjugate minimal pieces have been obtained by solving degree arguments, for example in [KPS], [Po], a trivial perturbation argument shows that constant mean curvature deformations of the constructed minimal surfaces also exist. Constant mean curvature one surfaces in H3 , also called Bryant ...

2 - Cambridge University Press

... The main aim of this article is to provide a detailed description and implementation of the algorithm for computing the Brauer–Manin obstruction of del Pezzo surfaces of degree 4. The algorithm is in general terms well known (see, for example, [9, 19]), but we shall build up a series of geometric an ...

... The main aim of this article is to provide a detailed description and implementation of the algorithm for computing the Brauer–Manin obstruction of del Pezzo surfaces of degree 4. The algorithm is in general terms well known (see, for example, [9, 19]), but we shall build up a series of geometric an ...

Minimal ideals and minimal idempotents

... all minimal left ideals of S; Theorem 6.7 states that if K(S) 6= ∅, i.e. if there is a minimal left ideal, then K(S) is the least two sided ideal of S. One might suspect that, by left-right symmetry, K(S) is also the union of all minimal right ideals of S. A moment’s reflection, however, shows that ...

... all minimal left ideals of S; Theorem 6.7 states that if K(S) 6= ∅, i.e. if there is a minimal left ideal, then K(S) is the least two sided ideal of S. One might suspect that, by left-right symmetry, K(S) is also the union of all minimal right ideals of S. A moment’s reflection, however, shows that ...

SYNTHETIC PROJECTIVE GEOMETRY

... However, it is clear that (a) and (a∗ ) are rephrasings of (1∗ ) and (1) respectively, and likewise (b) and (b∗ ) are rephrasings of (2∗ ) and (2) respectively. Thus (1) − (1 ∗ ) and (2) − (2∗ ) for (P ∗ , P ∗∗ ) are logically equivalent to (1) − (1 ∗ ) and (2) − (2∗ ) for (P, P ∗ ). As indicated a ...

... However, it is clear that (a) and (a∗ ) are rephrasings of (1∗ ) and (1) respectively, and likewise (b) and (b∗ ) are rephrasings of (2∗ ) and (2) respectively. Thus (1) − (1 ∗ ) and (2) − (2∗ ) for (P ∗ , P ∗∗ ) are logically equivalent to (1) − (1 ∗ ) and (2) − (2∗ ) for (P, P ∗ ). As indicated a ...

poincar ´e series of monomial rings with minimal taylor resolution

... that there exists a total ordering m1 ≺ m2 ≺ · · · ≺ mt on G(I) and a positive integer d ≤ t such that gcd(mi , . . . , mi+d−1 ) 6= 1 and gcd(mi , mi+d+ j ) = 1 for all i = 1, . . . ,t − d and j > 1. We use this description to compute the Hilbert series of H(K R ). Then Fröberg in [5] proved that P ...

... that there exists a total ordering m1 ≺ m2 ≺ · · · ≺ mt on G(I) and a positive integer d ≤ t such that gcd(mi , . . . , mi+d−1 ) 6= 1 and gcd(mi , mi+d+ j ) = 1 for all i = 1, . . . ,t − d and j > 1. We use this description to compute the Hilbert series of H(K R ). Then Fröberg in [5] proved that P ...

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 14

... We call a scheme of the form Proj S• (where S0 = A) a projective scheme over A, or a projective A-scheme. A quasiprojective A-scheme is an open subscheme of a projective A-scheme. The “A” is omitted if it is clear from the context; often A is some field. We now make a connection to classical termin ...

... We call a scheme of the form Proj S• (where S0 = A) a projective scheme over A, or a projective A-scheme. A quasiprojective A-scheme is an open subscheme of a projective A-scheme. The “A” is omitted if it is clear from the context; often A is some field. We now make a connection to classical termin ...

On function field Mordell-Lang: the semiabelian case and the

... some algebraically closed set A over which G is defined. Remark: We know that in the group G, because of finite U -rank, there are only a finite number of orthogonality classes of minimal types, hence we could work over some model M0 over which they are all represented; on the other hand, the argume ...

... some algebraically closed set A over which G is defined. Remark: We know that in the group G, because of finite U -rank, there are only a finite number of orthogonality classes of minimal types, hence we could work over some model M0 over which they are all represented; on the other hand, the argume ...

The Crust and the Ø-Skeleton: Combinatorial Curve Reconstruction

... computes the Delaunay triangulation of S 0 , and finally selects from the result all those edges whose two endpoints lie in S. For the β-skeleton, one computes the Delaunay triangulation of S and then selects each edge e for which the circumcircles of the adjacent triangles are centered on opposite ...

... computes the Delaunay triangulation of S 0 , and finally selects from the result all those edges whose two endpoints lie in S. For the β-skeleton, one computes the Delaunay triangulation of S and then selects each edge e for which the circumcircles of the adjacent triangles are centered on opposite ...

Publikationen - Mathematisches Institut

... Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, in: Annales de l'institut Fourier 65, 2015, H. 6, S. 2641-2680.Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, ...

... Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, in: Annales de l'institut Fourier 65, 2015, H. 6, S. 2641-2680.Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, ...

A Lefschetz hyperplane theorem with an assigned base point

... X is the quasi-projective variety Y n V . W is a Whitney stratification of Y such that V is a union of strata. A is a codimension 2 linear subspace of a fixed ambient projective space of Y . W jY nA is the Whitney stratification of Y n A obtained by restricting W . PA is the pencil of hyperplanes co ...

... X is the quasi-projective variety Y n V . W is a Whitney stratification of Y such that V is a union of strata. A is a codimension 2 linear subspace of a fixed ambient projective space of Y . W jY nA is the Whitney stratification of Y n A obtained by restricting W . PA is the pencil of hyperplanes co ...

Toroidal deformations and the homotopy type of Berkovich spaces

... Analytification of toric varieties Let X be a toric variety under the torus T, with open orbit X0 . Let S(X) denote the set of T1 -orbits in Xan (skeleton). The natural map rX : Xan → S(X) has a canonical section ...

... Analytification of toric varieties Let X be a toric variety under the torus T, with open orbit X0 . Let S(X) denote the set of T1 -orbits in Xan (skeleton). The natural map rX : Xan → S(X) has a canonical section ...

Problems on pencils of small genus

... Note that even a smooth fibre may be degenerate in terms of the way the canonical algebra is generated: e.g., a smooth hyperelliptic or trigonal fibre will mess up the generators and relations. History: The importance of hyperelliptic fibres for genus 3 pencils became clear in work of K. Konno and T ...

... Note that even a smooth fibre may be degenerate in terms of the way the canonical algebra is generated: e.g., a smooth hyperelliptic or trigonal fibre will mess up the generators and relations. History: The importance of hyperelliptic fibres for genus 3 pencils became clear in work of K. Konno and T ...

Dual Shattering Dimension

... First, Theorem 5.28 might fail, and the above conclusion might not hold (This is of course true). In real applications we use a much larger sample to guarantee that the probability of failure is so small that it can be practically ignored. A more serious issue is that Theorem 5.28 is defined o ...

... First, Theorem 5.28 might fail, and the above conclusion might not hold (This is of course true). In real applications we use a much larger sample to guarantee that the probability of failure is so small that it can be practically ignored. A more serious issue is that Theorem 5.28 is defined o ...

Arithmetic of hyperelliptic curves

... many points at infinity (i.e., points in C \ Caff ), and it is easy to check which of them are rational. So we can reduce the problem to that of determining Caff (Q). Now the set Q × Q of rational points in the affine plane is countable, so we can (in principle, at least) just enumerate all these po ...

... many points at infinity (i.e., points in C \ Caff ), and it is easy to check which of them are rational. So we can reduce the problem to that of determining Caff (Q). Now the set Q × Q of rational points in the affine plane is countable, so we can (in principle, at least) just enumerate all these po ...

a Gröbner Basis?

... need one more definition. Given a fixed ideal I in K[x1 , . . . , xn ] and a term order ≺ , a monomial a a xa = x11 · · · xnn is called standard if it is not in the initial ideal in≺ (I) . The number of standard monomials is finite if and only if every variable xi appears to some power in the initia ...

... need one more definition. Given a fixed ideal I in K[x1 , . . . , xn ] and a term order ≺ , a monomial a a xa = x11 · · · xnn is called standard if it is not in the initial ideal in≺ (I) . The number of standard monomials is finite if and only if every variable xi appears to some power in the initia ...

Uniform hyperbolicity of the curve graphs

... is related to, but simpler than, that used in [B]. In particular, it does not require the condition on moving centres (clause (2) of Proposition 3.1 of [B]) which complicated the argument there. Essentially the same statement can be found in Section 3.13 of [MS], though without a specific estimate f ...

... is related to, but simpler than, that used in [B]. In particular, it does not require the condition on moving centres (clause (2) of Proposition 3.1 of [B]) which complicated the argument there. Essentially the same statement can be found in Section 3.13 of [MS], though without a specific estimate f ...

Exact, Efficient, and Complete Arrangement Computation for Cubic

... Suppose we can compute gcds in a UFD R. Then we can also compute gcd(f, g) for non-zero f, g ∈ R[x]: Because of Corollary 2, gcd(f, g) is, up to a constant factor, the gcd of f, g regarded as elements of Q(R)[x], and we can compute that with the Euclidean Algorithm. Using Corollary 4, the error in t ...

... Suppose we can compute gcds in a UFD R. Then we can also compute gcd(f, g) for non-zero f, g ∈ R[x]: Because of Corollary 2, gcd(f, g) is, up to a constant factor, the gcd of f, g regarded as elements of Q(R)[x], and we can compute that with the Euclidean Algorithm. Using Corollary 4, the error in t ...

274 Curves on Surfaces, Lecture 5

... Proof. We first consider the case where all three lengths are equal to 0. This means that the horocycles are all tangent to each other as well as to the ideal boundary. The corresponding configuration of four circles is unique up to fractional linear transformations. Alternately, we can work with an ...

... Proof. We first consider the case where all three lengths are equal to 0. This means that the horocycles are all tangent to each other as well as to the ideal boundary. The corresponding configuration of four circles is unique up to fractional linear transformations. Alternately, we can work with an ...

Degrees of curves in abelian varieties

... Let (X, A) be a principally polarized abelian variety of dimension n defined over an algebraically closed field k. The degree of a curve C contained in X is d = C • X. The first question we are interested in is to find what numbers can be degrees of irreducibles curves C. When C generates X, we prov ...

... Let (X, A) be a principally polarized abelian variety of dimension n defined over an algebraically closed field k. The degree of a curve C contained in X is d = C • X. The first question we are interested in is to find what numbers can be degrees of irreducibles curves C. When C generates X, we prov ...

the usual castelnuovo s argument and special subhomaloidal

... birational map onto the image and subhomaloidal if the (closure of a) general �ber of the associated rational map is a linear projective space. It is said special if the base locus scheme of the linear system is a smooth irreducible subvariety X ⊂ Pr . Special (sub)homaloidal systems of quadrics des ...

... birational map onto the image and subhomaloidal if the (closure of a) general �ber of the associated rational map is a linear projective space. It is said special if the base locus scheme of the linear system is a smooth irreducible subvariety X ⊂ Pr . Special (sub)homaloidal systems of quadrics des ...

MONOMIAL RESOLUTIONS Dave Bayer Irena Peeva Bernd

... Let M be a monomial ideal in the polynomial ring S = k[x1 , . . . , xn ] over a ﬁeld k. We are interested in the problem, ﬁrst posed by Kaplansky in the early 1960’s, of ﬁnding a minimal free resolution of S/M over S. The diﬃculty of this problem is reﬂected in the fact that the homology of arbitrar ...

... Let M be a monomial ideal in the polynomial ring S = k[x1 , . . . , xn ] over a ﬁeld k. We are interested in the problem, ﬁrst posed by Kaplansky in the early 1960’s, of ﬁnding a minimal free resolution of S/M over S. The diﬃculty of this problem is reﬂected in the fact that the homology of arbitrar ...

Universal unramified cohomology of cubic fourfolds containing a plane

... from P . Its discriminant divisor D ⊂ P is a sextic curve. The rationality of a cubic fourfold containing a plane over C is also a well known problem. Expectation. The very general cubic fourfold containing a plane over C is irrational. Assuming that the discriminant divisor D is smooth, the discrim ...

... from P . Its discriminant divisor D ⊂ P is a sextic curve. The rationality of a cubic fourfold containing a plane over C is also a well known problem. Expectation. The very general cubic fourfold containing a plane over C is irrational. Assuming that the discriminant divisor D is smooth, the discrim ...