scholarly journals Tangent cones to generalised theta divisors and generic injectivity of the theta map

2017 ◽  
Vol 153 (12) ◽  
pp. 2643-2657 ◽  
Author(s):  
George H. Hitching ◽  
Michael Hoff
Keyword(s):  
Vector Bundle ◽  
Tangent Cone ◽  
Constructive Proof ◽  
Tangent Cones ◽  
Stable Vector Bundle ◽  
Theta Divisor ◽  
General Curve

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^{0}(C,E)=r+1$. For $g\geqslant (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the generalised theta divisor $\unicode[STIX]{x1D6E9}_{E}$ at ${\mathcal{O}}_{C}$. We use this to give a constructive proof and a sharpening of Brivio and Verra’s theorem that the theta map $\mathit{SU}_{C}(r){\dashrightarrow}|r\unicode[STIX]{x1D6E9}|$ is generically injective for large values of $g$.

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 LIFTINGS OF A MONOMIAL CURVE

2018 ◽  
Vol 98 (2) ◽  
pp. 230-238
Author(s):  
MESUT ŞAHİN
Keyword(s):  
Tangent Cone ◽  
Tangent Cones ◽  
Monomial Curves ◽  
Original Curve ◽  
Monomial Curve

We study an operation, that we call lifting, creating nonisomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen–Macaulay tangent cones even if the tangent cone of the original curve is not Cohen–Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen–Macaulay tangent cones when the original monomial curve has a Cohen–Macaulay tangent cone. In this case, all the Betti sequences are just the Betti sequence of the original curve.

ON MAXIMALLY FROBENIUS DESTABILISED VECTOR BUNDLES

2019 ◽  
Vol 99 (2) ◽  
pp. 195-202
Author(s):  
LINGGUANG LI
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Closed Field ◽  
Projective Curve ◽  
Stable Vector Bundle ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

scholarly journals Stable Higgs bundles over positive principal elliptic fibrations

2018 ◽  
Vol 5 (1) ◽  
pp. 195-201
Author(s):  
Indranil Biswas ◽  
Mahan Mj ◽  
Misha Verbitsky
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Complex Manifold ◽  
Higgs Bundle ◽  
Compact Complex Manifold ◽  
Holomorphic Line Bundle ◽  
Higgs Bundles ◽  
Stable Vector Bundle ◽  
Hermitian Metric ◽  
The Third

AbstractLet M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B0, where B0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L ⊕ Π ⃰B0, Π ⃰ ɸX), where (B0, ɸX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

scholarly journals Ample vector bundles on a rational surface (higher rank)

1977 ◽  
Vol 66 ◽  
pp. 77-88
Author(s):  
Toshio Hosoh
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Rational Surface ◽  
Sufficient Condition ◽  
Chern Classes ◽  
Stable Vector Bundle ◽  
Rank 2 ◽  
Higher Rank

In the previous paper [1], we showed that the set of simple vector bundles of rank 2 on a rational surface with fixed Chern classes is bounded and we gave a sufficient condition for an H-stable vector bundle of rank 2 on a rational surface to be ample. In this paper, we shall extend the results of [1] to the case of higher rank.

scholarly journals Differential inclusions with state constraints

1989 ◽  
Vol 32 (1) ◽  
pp. 81-98 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Inclusion ◽  
Tangent Cone ◽  
Differential Inclusions ◽  
State Constraints ◽  
Tangent Cones ◽  
Viability Theory ◽  
Constraint Set

In “Viability Theory”, we select trajectories which are viable in the sense that they always satisfy a given constraint. Since the fundamental work of Nagumo [26], we know that in order to guarantee existence of viable trajectories, we need to satisfy certain tangential conditions. In the case of differential inclusions and using the modern terminology and notation of tangent cones, this condition takes the form F(t, x) ∩ TK#φ, where F(.,.) is the orientor field involved in the differential inclusion, K is the viability (constraint) set and TK(x) is the tangent cone to K at x. Results on the existence of viable solutions for differential inclusions can be found in Aubin–Cellina [2] and Papageorgiou [30,32].

scholarly journals Optimality Conditions for Group Sparse Constrained Optimization Problems

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 84
Author(s):  
Wenying Wu ◽  
Dingtao Peng
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Stationary Point ◽  
Tangent Cone ◽  
Optimization Problems ◽  
Stationary Points ◽  
Tangent Cones ◽  
Sufficient Optimality Conditions ◽  
Normal Cones ◽  
The Relationship

In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones and normal cones, four types of stationary points for GSCO problems are given: TB-stationary point, NB-stationary point, TC-stationary point and NC-stationary point, which are used to characterize first-order optimality conditions for GSCO problems. Furthermore, both the relationship among the four types of stationary points and the relationship between stationary points and local minimizers are discussed. Finally, second-order necessary and sufficient optimality conditions for GSCO problems are provided.

scholarly journals Ample vector bundles on a rational surface

1975 ◽  
Vol 59 ◽  
pp. 135-148 ◽  
Author(s):  
Toshio Hosoh
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Rational Surface ◽  
Ruled Surface ◽  
Necessary Condition ◽  
Abelian Surface ◽  
Closed Field ◽  
Stable Vector Bundle ◽  
Singular Curve ◽  
Rank 2

On a complete non-singular curve defined over the complex number field C, a stable vector bundle is ample if and only if its degree is positive [3]. On a surface, the notion of the H-stability was introduced by F. Takemoto [8] (see § 1). We have a simple numerical sufficient condition for an H-stable vector bundle on a surface S defined over C to be ample; let E be an H-stable vector bundle of rank 2 on S with Δ(E) = c1(E)2 - 4c2(E) ≧ 0, then E is ample if and only if c1(E) > 0 and c2(E) > 0, provided S is an abelian surface, a ruled surface or a hyper-elliptic surface [9]. But the assumption above concerning Δ(E) evidently seems too strong. In this paper, we restrict ourselves to the projective plane P2 and a rational ruled surface Σn defined over an algebraically closed field k of arbitrary characteristic. We shall prove a finer assertion than that of [9] for an H-stable vector bundle of rank 2 to be ample (Theorem 1 and Theorem 3). Examples show that our result is best possible though it is not a necessary condition (see Remark (1) §2).

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