open problems in algebraic geometry

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Resolution of singularities in characteristic p. Evidences on Hartshorne's conjecture? In that process, the search for finding the “true” nature of the problem at hand is the impetus for our thoughts. for a discussion including some references. Seattle 2005. I think?). 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. Griffiths conjecture is also known to be true for general vector bundles on curves! To learn more, see our tips on writing great answers. The weak conjecture fails for $n=3$ and $4$ -- there are examples (due to Horrocks and Mumford) of non-split vector bundles of rank 2 on $\mathbf{P}^4_{\mathbf{C}}$, but so far as I know the question if any such examples exist for $n>4$ is open. Introduction 1 2. This question was lucky enough that Richard Borcherds offered a very nice answer and potentially there will be further answers that we can enjoy and ultimately this will be a useful source. Open problems/questions in representation theory and around? References 2 1.1. Mathematical books usually deal with fully developed theories. In its weak form it says that any rank 2 vector bundle on $\mathbf{P}^n_{\mathbf{C}},n>6$ is a direct sum of line bundles, which implies that any codimension 2 smooth subvariety whose canonical class is a multiple of the hyperplane sectionis a complete intersection. Then the cycle class map $$\mathrm{CH}^r(X) \otimes_\mathbf{Z} \mathbf{Q}_\ell \to \mathrm{H}^{2r}(\bar{X},\mathbf{Q}_\ell(r))^{G_k}$$ is surjective. Her answer was equivocal, if memory serves me right. Open problems in Birational Geometry, after BCHM, Open Problems in Algebraic Topology and Homotopy Theory. No. Joe Harris had some slides a few years ago with regards to this

As of my comment, this question currently has four votes to close as "off topic", but it's certainly not that, it's just too vague. It also might even be true in characteristic $p > 0$. I would be interested to see some of the answers. For vector bundles, a longstanding open problem is the classification of vector bundles over projective spaces. there exists a birational automorphism of $\mathbb{P}^2$ which transforms the curve into a line.

Then. Hartshorne has said that his entire mathematical career was structured around trying to prove the set-theoretic intersection conjecture (I.2.17d), and it still seems to be open. rev 2020.10.7.37757, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us.
site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. MathOverflow is a question and answer site for professional mathematicians. What are some open problems in toric varieties? Browse other questions tagged ag.algebraic-geometry open-problems or ask your own question. Do you mean open problems mentioned in Hartshorne's book or conjectures that he made? Are there other problems that I missed? EDIT: The Maximal Rank Conjecture was proved by Eric Larson in his PhD thesis; see: https://arxiv.org/abs/1711.04906. This is the gist of the present volume. Also, if either were solved, I imagine people would talk. If "too broad/vague" were a criterion on the list of reasons to close, I would vote to close. 3 2.1. tools from algorithmic semi-algebraic geometry, as well as algebraic topology, which make these advances possible. One of them is atonishingly simple but still completely open : Let $E$ a rank $2$ vector bundle on $\mathbb{P}^n$, with $n \geq 7$. Semi-algebraic Geometry: Background 3 3. There are examples of indecomposable rank $2$ vector bundles on $\mathbb{P}^5$ in characteristic $2$ due to Tango and Kumar-Peterson-Rao (independently). See http://www.math.harvard.edu/~chaoli/doc/TateConjecture.html. What are some open problems in algebraic geometry?

Updates to Stanley's 1999 survey of positivity problems in algebraic combinatorics? Cancellation Conjecture. We can also mention two other major open problems : The abundance conjecture, stating that if a $K_X+\Delta$ is klt and nef, then it is semi-ample (a multiple has no base-point), The Griffith's conjecture : if $E$ is an ample vector bundle over a compact complex manifold, then it is Griffith-positive. See here Complex vector bundles that are not holomorphic for some more information. problems mentioned in hartshornes book (the ** excercises). Exercises 1: categorical preliminaries 6 3. Lecture 1: What is algebraic geometry? I wish the second reference contained some problems over the complex numbers. Also there are many refinements (and generalizations) of this conjecture. Thanks, Mahdi, that's interesting.

Books 2 1.2. Topological Preliminaries 22 6. But here we present work at an earlier stage—when challenging questions can give new directions to mathematical research. It's also often stated in the complex analytic world. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The overall consensus (that's too strong a word ... plurality opinion?) I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory. If $X\times \mathbb{C}\cong \mathbb{C}^{m+1}$ then $X\cong \mathbb{C}^m$. Please enable Javascript in your web browser. This is analogous to the Hodge conjecture for complex varieties. (Added later) A very old major problem is that of finding which moduli spaces of curves are unirational. Only part 1 is known in low dimension - part 2 is open even in dimension 3. Asking for help, clarification, or responding to other answers. Open problems in Birational Geometry, after BCHM. There's also the big open question (I think it's still open) about whether rationally connected varieties are always unirational. There are a lot of open problems and conjectures in K-theory, which are "sometimes" inspired by linear algebra.

Let me mention a couple of problems related to vector bundles on projective spaces. Motives and Algebraic cycles: A selection of conjectures and open questions, Joseph Ayoub. The answers can give prople some clues for what to look for in the ICM talks and bulletin articles Mathew referred to. Swapping out our Syntax Highlighter, Responding to the Lavender Letter and commitments moving forward, The most outrageous (or ridiculous) conjectures in mathematics. *Proving finite generation of the canonical ring for general type used to be open though I think it was recently solved; I'm not sure about the details. Good introductory references on moduli (stacks), for arithmetic objects, Applications of derived categories to “Traditional Algebraic Geometry”.
Making statements based on opinion; back them up with references or personal experience. It is e.g. Is there a precise relationship between the goals of moduli theory and the minimal model program? A rational cuspidal curve in $\mathbb{P}^2$ is rectifiable, i.e.

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