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MA4A5 Algebraic Geometry Support Classes

Week 2: We discussed some of the basics of projective space. If you did not feel comfortable with the material the class was structured off these notes which were taught as a short course on projective geometry at the University of Oxford. These are not my own notes, if you find them useful please thank Balazs.

If one googles projective geometry Oxford, there are also exercise sheets with easier questions which may prove of use. Similarly, the geometry course at Warwick will have exercises which may be of use.

Week 3: We discussed sheaves and quasi-projective varieties at length; in particular why we care about regular and rational functions. We also proved that every non-empty Zariski open subset of an irreducible space is dense.

Week 4: We discussed rational maps and morphisms and the differences between the two. In particular we focused on asking whether certain rational maps could be extended to projective morphisms and attempted some exam questions from the 2016 paper.

Week 5: Segre and Veronese by Alice Cuzzucoli.

Week 6: Review of Grassmanians, projective and proper morphisms. For those students struggling to understand a concept in the lecture notes, an alternative one could try is these notes by Andreas Gathmann. There is also a follow on from these which covers quasi-coherent sheaves, cohomology, intersection theory, Chern classes etc.

Week 7: A question about Zariski dense arguments (which may help clarify injectivity of the restriction map in Assignment 2 Question 3). Functions vanishing on a dense subset vanish on the closure. Images of morphisms, when X affine the image may not be closed, open or could even be a union of both. Transcendence degree and computing dimension. In the course regular functions are defined first and then rational functions. This corresponds scheme theoretically to the way one often views the rational function field as the stalk of the sheaf of regular functions at the generic point. Less technically however one can think in this way: for $X$ closed affine, consider polynomial functions on $X$ in the k-algebra $k[X]$. This is defined to be $K[\mathbb{A}^n]/I(X)$. One can define a regularity condition at every point (that the rational function- the quotient of 2 polynomial functions is defined at that point) and $\mathcal{O}_X(U)$ is then the rational functions regular for every point in $U$. Then the global sections of this sheaf will precisely be the rational functions defined on all of $X$, which will precisely be $K[X]$. Since $K[X]$ will not necessarily be a UFD, one has many different representatives possibly for such a regular function. To see how this can be done you can look at page 77 and page 125 here regular and rational functions.

Week 8: Upper semi-continuity of fibre dimension and the final 8 mark question from this exam. I did not finish this but motivated the usefulness of incidence varieties (local form of blow-up). $\mathbb{P}^1 \times \mathbb{P}^1$ birational to $\mathbb{P}^2$ but not isomorphic. $\mathbb{A}^1 \times \mathbb{P}^1$ neither affine nor projective. Uses of the fact there are no-non constant morphisms from $\mathbb{P}^n \to \mathbb{P}^m, m<n$.

Week 9: N/A

Week 10: The classification problem, discrete invariants and moduli. Chow varieties, Cayley forms and degree.

As a supplement to these support classes, I will hold a revision session in the week prior to your exam where we can cover exam style problems.