## An indecomposable subbundle of a direct sum

Can one have $$N \subset L_1 \oplus L_2$$ where $$N$$ is an indecomposable vector bundle of rank $$2$$ and degree $$n$$ on a smooth projective algebraic curve $$X$$ and $$L_1, L_2$$ are line bundles on $$X$$ such that the sum of degrees of $$L_1, L_2$$ is $$n+1$$?

## An indecomposable subbundle of a direct sum

Can one have $$N \subset L_1 \oplus L_2$$ where $$N$$ is an indecomposable vector bundle of rank $$2$$ and degree $$n$$ on a smooth projective algebraic curve $$X$$ and $$L_1, L_2$$ are line bundles on $$X$$ such that the sum of degrees of $$L_1, L_2$$ is $$n+1$$?

## Do negative indecomposable bundles on curves have sections?

Let $$X$$ be a smooth projective curve, and $$E$$ an indecomposable vector bundle on $$X$$ with $$\mathrm{deg} E<0$$. Is it true that $$H^0(X,E)=0$$?

This is true if $$E$$ is a line bundle, which means it is also true whenever $$X$$ is $$\mathbb{P}^1$$, since all vector bundles split here.

It is also true by results of Atiyah if $$X$$ is an elliptic curve. What about for curves of higher genus?

The assumption that $$E$$ is indecomposable is of course necessary.

## Categories with every indecomposable object being uniserial

Let $$A$$ be an abelian category. A Jordan-Hölder series for an object $$X$$ is a filtration $$0 such that $$X_i/X_{i-1}$$ are simple. Call $$X$$ uniserial in case it has a unique Jordan-Hölder series up to isomorphism.

Questions:

1. Is there a name for such $$A$$ with the property that every indecomposable object is uniserial? Have they been studied in this generality? For Artin algebras those are exactly the Nakayama algebras.

2. Is there a classification of such abelian categories with every indecomposable object being uniserial and such that the abelian category has global dimension equal to one? For finite dimensional algebras those are exactly direct products of matrix rings over division fields and upper triangular matrix rings over division rings. Are there easy examples that are not of this form?

3. Is there even a general classification for abelian categories such that any indecomposable object is uniserial?

## Is every indecomposable homogeneous continuum unicoherent?

• Continuum = compact connected metrizable space

• Indecomposable = not the union of any two proper subcontinua.

• Homogeneous = for every two points $$x$$ and $$y$$ there is a homeomorphism of the space onto itself which maps $$x$$ to $$y$$.

• Unicoherent = the intersection of every two subcontinua is connected.

Examples of indecomposable homogeneous continua include solenoids, the pseudoarc, and solenoids of pseudoarcs (a solenoid with each point is blown up into a pseudoarc). I think it’s an open problem to determine whether this list is complete, but so far all of the examples are unicoherent. Is there a theorem stating that such continua must be unicoherent?