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Tuesday, April 17, 2018

**Abstract:** Let $K$ be an algebraically closed field and let $V$ be a projective variety (reduced, irreducible, nondegenerate) embedded in $\mathbb{P}^n_K$. There are two notions of degree associated to $V$: (1) the degrees of the generators of the defining homogeneous ideal of $V$, and (2) the number of points of intersection between $V$ and a linear space of complementary dimension. In the plane these two notions agree, but in higher dimension there was a only the conjectured inequality $\mathrm{maxdeg}(P) \leq \deg(V(P))$ for a prime ideal $P$. Recent joint work with Irena Peeva produced counterexamples to this conjecture and the Eisenbud-Goto conjecture. However, using the recent solution to Stillman's Conjecture by Ananyan and Hochster, we show that there is a bound on the defining equaltions of any variety purely in terms of its degree. This is joint work with Caviglia, Chardin, Peeva, and Varbaro.