scholarly journals On some Siegel threefold related to the tangent cone of the Fermat quartic surface

2017 ◽  
Vol 21 (3) ◽  
pp. 585-630 ◽  
Author(s):  
Takeo Okazaki ◽  
Takuya Yamauchi
Keyword(s):  
Tangent Cone ◽  
Quartic Surface

scholarly journals Two algebraic deformations of a K3 surface

1981 ◽  
Vol 82 ◽  
pp. 1-26
Author(s):  
Daniel Comenetz
Keyword(s):  
Hyperelliptic Curve ◽  
Double Cover ◽  
Rational Curves ◽  
K3 Surface ◽  
Quartic Surface ◽  
Quadric Cone ◽  
Birational Model ◽  
Affine Line ◽  
Characteristic 2 ◽  
General Member

Let X be a nonsingular algebraic K3 surface carrying a nonsingular hyperelliptic curve of genus 3 and no rational curves. Our purpose is to study two algebraic deformations of X, viz. one specialization and one generalization. We assume the characteristic ≠ 2. The generalization of X is a nonsingular quartic surface Q in P3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the affine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P3.

The minimal resolution conjecture on a general quartic surface in P3

2019 ◽  
Vol 223 (4) ◽  
pp. 1456-1471
Author(s):  
M. Boij ◽  
J. Migliore ◽  
R.M. Miró-Roig ◽  
U. Nagel
Keyword(s):  
Minimal Resolution ◽  
Quartic Surface

scholarly journals Complete Intersection Monomial Curves and the Cohen—Macaulayness of Their Tangent Cones

2019 ◽  
Vol 26 (04) ◽  
pp. 629-642
Author(s):  
Anargyros Katsabekis
Keyword(s):  
Complete Intersection ◽  
Tangent Cone ◽  
Affine Space ◽  
Intersection Property ◽  
Tangent Cones ◽  
Monomial Curves ◽  
Monomial Curve

Let C(n) be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve C(n + wv), where w > 0 is an integer and v ∈ ℕ4. In addition, we investigate the Cohen–Macaulayness of the tangent cone of C(n + wv).

scholarly journals On the Cohen-Macaulay Property for Quadratic Tangent Cones

10.37236/5793 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Dumitru I. Stamate
Keyword(s):  
Tangent Cone ◽  
Gröbner Basis ◽  
Numerical Semigroup ◽  
Tangent Cones ◽  
Grobner Basis

Let $H$ be an $n$-generated numerical semigroup such that its tangent cone $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadratic relations. We show that if $n<5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Cohen-Macaulay, and for $n=5$ we explicitly describe the semigroups $H$ such that $\operatorname{gr}_\mathfrak{m} K[H]$ is not Cohen-Macaulay. As an application we show that if the field $K$ is algebraically closed and of characteristic different from two, and $n\leq 5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Koszul if and only if (possibly after a change of coordinates) its defining ideal has a quadratic Gröbner basis.

scholarly journals VECTOR BUNDLES, DUALITIES AND CLASSICAL GEOMETRY ON A CURVE OF GENUS TWO

2007 ◽  
Vol 18 (05) ◽  
pp. 535-558 ◽  
Author(s):  
QUANG MINH NGUYEN
Keyword(s):  
Vector Bundles ◽  
Double Cover ◽  
Line Bundles ◽  
Theta Divisor ◽  
Quartic Surface ◽  
Stable Vector Bundles ◽  
Symmetric Configuration ◽  
Classical Geometry ◽  
Genus Two ◽  
Linear Sections

Let C be a curve of genus two. We denote by [Formula: see text] the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by Jd the variety of line bundles of degree d on C. In particular, J1 has a canonical theta divisor Θ. The space [Formula: see text] is a double cover of ℙ8 = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual [Formula: see text], where J1 is embedded, there is a unique cubic hypersurface singular along J1, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.

A Cyclic Involution of Period Eleven

1951 ◽  
Vol 3 ◽  
pp. 155-158
Author(s):  
W. R. Hutcherson
Keyword(s):  
Quartic Surface ◽  
Cubic Surfaces

In two earlier papers * the writer discussed involutions of periods five and seven on certain cubic surfaces in S3. In this paper, a quartic surface containing a cyclic involution of period eleven is considered. The surface

Ordinary Singularities of Algebraic Curves

1981 ◽  
Vol 24 (4) ◽  
pp. 423-431 ◽  
Author(s):  
Ferruccio Orecchia
Keyword(s):  
Singular Point ◽  
Vector Space ◽  
Local Ring ◽  
Maximal Ideal ◽  
Tangent Cone ◽  
Algebraic Curves ◽  
Vector Spaces ◽  
Generic Position

AbstractLet A be the local ring at a singular point p of an algebraic reduced curve. Let M (resp. Ml,..., Mh) be the maximal ideal of A (resp. of Ā). In this paper we want to classify ordinary singularities p with reduced tangent cone: Spec(G(A)). We prove that G(A) is reduced if and only if: p is an ordinary singularity, and the vector spaces span the vector space . If the points of the projectivized tangent cone Proj(G(A)) are in generic position then p is an ordinary singularity if and only if G(A) is reduced. We give an example which shows that the preceding equivalence is not true in general.

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