What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over projective varieties/curves.
What are the open big problems in algebraic geometry and vector bundles? More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over projective varieties/curves. 


A few of the more obvious ones: * Resolution of singularities in characteristic p For vector bundles, a longstanding open problem is the classification of vector bundles over projective spaces. (Added later) A very old major problem is that of finding which moduli spaces of curves are unirational. It is classical that the moduli space is unirational for genus at most 10, and I think this has more recently been pushed to genus about 13. Mumford and Harris showed that it is of general type for genus at least 24. As far as I know most of the remaining cases are still open. 


Let me mention a couple of problems related to vector bundles on projective spaces.






We can also mention two other major open problems :



There's also the big open question (I think it's still open) about whether rationally connected varieties are always unirational. I think people believe the answer is NO, but they don't know an example. Joe Harris had some slides a few years ago with regards to this Seattle 2005 


There's also Fujita's conjecture. Conjecture: Suppose $X$ is a smooth projective dimensional complex algebraic variety with ample divisor $A$. Then
It's also often stated in the complex analytic world. Also there are many refinements (and generalizations) of this conjecture. For example, the assumption that $X$ is smooth is probably more than you need (something close to rational singularities should be ok). It also might even be true in characteristic $p > 0$. It's known in relatively low dimensions (up to 5 in case 1. I think?) 


Linearization Conjecture. Every algebraic action of $\mathbb{C}^*$ on $\mathbb{C}^n$ is linear in some coordinates of $\mathbb{C}^n$. Open for $n>3$. Cancellation Conjecture. If $X\times \mathbb{C}\cong \mathbb{C}^{m+1}$ then $X\cong \mathbb{C}^m$. Open for $m>2$. CoolidgeNagata Conjecture. A rational cuspidal curve in $\mathbb{P}^2$ is rectifiable, i.e. there exists a birational automorphism of $\mathbb{P}^2$ which transforms the curve into a line. 


In connection to vector bundles over $\mathbb{P}^n$, Hartshorne's paper from 1979 provides a list of open problems. The paper is "Algebraic vector bundles on projective spaces: A problem list" Topology, 18:117–128, 1979. I don't know which of those problems are still open, but I would be interested in knowing how much progress has been made on those problems, since 1979. 

