Classification of higher dimensional algebraic varietiesģ. Their singularities were recently classified and resolved by Cano.Ģ. Finally one could try to transfer these statements to higher-dimensional cases, for example on foliations of the 3-dimensional complex space by 2-dimensional leaves. The model of such general results is Mustataís recent characterization of ordinary singularities of special types by jets and arcs. The results of these computations should help to predict statements for the general case. On examples this can be done by using computer algebra programs relying on Groebner basis techniques, like MACAULAY or in simpler cases, MAPLE. The PhD project should investigate the existing notions of good plane foliation singularities by means of jets and arcs tangent to the foliation. Jets to singularities of complex foliations Current PhD opportunities in the Math Department (different deadlines) >įoliation singularities, higher dimensional algebraic geometry, K3 surfaces, algebraic cycles and deformation theory, moduli spaces of curves and Gromov-Witten theory.ġ.Mathematical Sciences Health and Safety Intranet.Institute for Financial and Actuarial Mathematics.If one thinks about this, we are basically "formally taking a point with transcendental coefficients. The point is that we can define "smooth at a point" in a way that makes sense for "normal points" and generic points, so we can literally check that the generic point is smooth by plugging in equations. For example, let us say that we want to prove that a variety of dimension $n$ is generically smooth. I think I can at least convince you that it's a good idea to work with non-reduced rings, which aren't really captured by their maximal ideals. (Of course, serious such studies were made by the Italian geometers, by Lefschetz, by Igusa, by Shimura, and by many others before Grothendieck's invention of schemes, but the whole point of schemes is to clarify what came before and to give a precise and workable theory that encompasses all of the contexts considered in the "old days", and is also more systematic and more powerful than the older techniques.) Of varieties) requires scheme-theoretic techniques and the consideration of non-closed points. schemes over $\mathbb Z$, or geometric families, i.e. What is the upshot? Basically, any serious study of varieties in families (whether arithmetic families, i.e. This is completelyĪnalogous to the situation considered in my linked answer, of taking integral solutions to a a Diophantine equation and then reducing them mod $p$. Homomorphism $\mathbb C \to \mathbb C$ given by $t \mapsto t_0$ (specialization at $t_0$). Then you need to pass from $\mathbb C(t)$ to $\mathbb C$, so that you can apply the But suppose you want to study the geometry for one particular value of $t_0$ of $t$. To study the generic behaviour of this equation, you can think of it as a variety over $\mathbb C(t)$. Well, suppose you have an equation (like $y^2 = x^3 t$) which you want to study, where you think of $t$ as a parameter. The ring $\mathbb C$ behaves much like $\mathbb Z$, and so one can have the same discussion with $\mathbb Z$ and $\mathbb Q$ replaced by $\mathbb C$ and $\mathbb C(t)$. These examples may give impression that non-closed points are most important in arithmetic situations, but actually that is not the case. $\mathbb Z \to \mathbb Q$, the preimage of maximal ideals are prime, but not maximal. In terms of rings (and connecting to Qiaochu's answer), under the natural map See my answer here for a brief discussion of how points that are closed in one optic (rational solutions to a Diophantine equation, which are closed points on the variety over $\mathbb Q$ attached to the Diophantine equation) become non-closed in another optic (when we clear denominators and think of the Diophantine equation as defining a scheme over $\mathbb Z$).
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