Description

Speaker: Amit Kumar Singh.

Affiliation: IIT Madras.

Date and Time: Thursday 24 October, 11:45 am - 12:45 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Semi-stability of certain vector bundles on elliptic curves.

Abstract: Abstract Let L be a line bundle of degree d on an elliptic curve C and ϕ : C → P

n

is a morphism

given by a sub-linear system of the complete linear system |L| of dimension n + 1. When d = 4, n

= 2, we prove that ϕ

∗TPn is semi-stable if deg(ϕ(C)) > 1. Moreover, we prove that ϕ

∗TPn is isomorphic to direct sum of two isomorphic line bundles if and only if deg(ϕ(C)) = 2. Conversely, for any

rank two semi-stable vector bundle E on an elliptic curve C of degree 4, there is a non-degenerate

morphism ϕ :C → P

n

such that ϕ

∗TPn (−1) = E. More precisely, E is isomorphic to direct sum of two

isomorphic line bundles if and only if deg(ϕ(C)) = 2. Further E is either indecomposable or direct

sum of non-isomorphic line bundles if and only if deg(ϕ(C)) = 4. When d = 5, n = 3, we compute

the Harder-Narasimhan filtration of ϕ

∗TPn .

Affiliation: IIT Madras.

Date and Time: Thursday 24 October, 11:45 am - 12:45 pm.

Venue: Ramanujan Hall, Department of Mathematics.

Title: Semi-stability of certain vector bundles on elliptic curves.

Abstract: Abstract Let L be a line bundle of degree d on an elliptic curve C and ϕ : C → P

n

is a morphism

given by a sub-linear system of the complete linear system |L| of dimension n + 1. When d = 4, n

= 2, we prove that ϕ

∗TPn is semi-stable if deg(ϕ(C)) > 1. Moreover, we prove that ϕ

∗TPn is isomorphic to direct sum of two isomorphic line bundles if and only if deg(ϕ(C)) = 2. Conversely, for any

rank two semi-stable vector bundle E on an elliptic curve C of degree 4, there is a non-degenerate

morphism ϕ :C → P

n

such that ϕ

∗TPn (−1) = E. More precisely, E is isomorphic to direct sum of two

isomorphic line bundles if and only if deg(ϕ(C)) = 2. Further E is either indecomposable or direct

sum of non-isomorphic line bundles if and only if deg(ϕ(C)) = 4. When d = 5, n = 3, we compute

the Harder-Narasimhan filtration of ϕ

∗TPn .

Description

Ramanujan Hall, Department of Mathematics

Date

Thu, October 24, 2019

Start Time

11:45am-12:45pm IST

Duration

1 hour

Priority

5-Medium

Access

Public

Created by

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Updated

Tue, October 22, 2019 12:40pm IST