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Wednesday, December 6, 2017

**Abstract:** A variety $X$ is ruled if there is a bi-rational map $\mathbb{P}^1 × Y \rightarrow X$ for some variety Y . The study of ruled varieties is motivated by the desire to study rational varieties by weakening the condition. If a variety $X$ has a $\mathbb{G}_a$-action, then one sees that it is ruled. Also, simplicial toric varieties often have non-reductive automorphism groups. An example is $\mathbb{P}(1 : 1 : 2)$ whose automorphism group is isomorphic to $GL(2, k)\times \mathbb{G}_a^3$(semi-direct product). As a result one finds that constructing a type of non-reductive GIT is a worthy venture. If one wants to construct this type of non-reductive GIT one first needs to understand actions of $\mathbb{G}_a$, the simplest non-reductive group. The earliest theorem regarding Ga was Weitzenb¨ock’s Theorem, which states that a representation $\mu : \mathbb{G}_a → GL(V_n)$ has a finitely generated ring of invariants $k[X]^{\mathbb{G}_a}$ if the field $k$ is an algebraically close field of characteristic zero. Work then focused on whether the invariant ring of non-linear actions of Ga is finitely generated. The answer is no. At this point people wished to answer the Weitzenbock conjecture, namely whether the result of Weitzenb¨ock’s Theorem still holds if the characteristic of the field is positive. It is this question we wish to answer in the following talks. In this talk we begin the process of producing a representation $\mu : \mathbb{G}_a → GL(V_6)$ over an algebraically closed field k of characteristic $p > 2$ such that the ring of invariants $k[x_1,\ldots,x_6]^{\mathbb{G}_a}$ is not a finitely generated k-algebra. In order to do this, we reduce the problem of finding a counterexample to the Weitzenbock conjecture to a curve counting problem, and then use this reduction to further reduce this problem to a problem about the support of a bi-graded ring.