3 problems

Let $R$ be a complete local ring, and let $F$ be a short complex in the perfect-complex category, meaning a short complex in $\operatorname{perf} R$ supported on the maximal ideal.…

Let $(R,\mathfrak m)$ be a local ring, let $e(R)$ denote its Hilbert–Samuel multiplicity, and let $F$ be a short complex supported on the maximal ideal, meaning a non-exact complex…

Let $x_1,\dots,x_r\in {\mathbb P}^2$ be very general points, and let $D$ be an effective divisor in ${\mathbb P}^2$. The stronger bound conjecture. The degree and assigned multipli…