an open source textbook and reference work on algebraic geometry

We construct an explicit smooth projective Fano fourfold $X$ over an algebraically closed field $k$ of characteristic two with $H^1(X,\mathcal{O}_X)\simeq k$. Moreover, $X$ does not admit any free…

For an isolated hypersurface singularity f:(\CC^n,0_\rightarrow(\CC,0), we study the Łojasiewicz exponent ł(f) of f. We formulate three conjectures concerning: (i) attainment of ł(f) on coordinate polar curves, (ii) topological invariance of ł(f), and (iii) a local Newton-diagram formula for ł(f) in the Kushnirenko non-degenerate case. We compute ł(f) for all singularities of modality at most 2 in Arnold’s list and for several further families, obtaining systematic evidence in favor of these conjectures. We also find to what extent the collection of numerical invariants {modality⁡(f),ł(f),\mu(f),corank⁡(f),stabord⁡(f)} determines the stable equivalence class of f for the aforementioned singularities.

Kawamata bounds for Fano threefolds and the Graded Ring Database - Volume 261

We construct a polarized smooth projective fivefold and prove that it is K-polystable but does not admit a constant scalar curvature Kähler metric. This disproves the Yau--Tian--Donaldson conjecture f...

I am a PhD student at the London School of Geometry and Number Theory (LSGNT) and recently passed my PhD defense with minor corrections. My PhD supervisor is Alessio Corti from Imperial College…

I am interested in algebraic geometry and machine learning. I like to think about how computers, and AI, through respective algorithm and architecture design, can be used to help classify objects in birational geometry. Currently, I am working on understanding Fano 4-fold hypersurfaces, and how deep reinforcement learning can be used to help enumerate them.

A mathematician working at Anthropic says he used the AI model Claude Fable 5 to uncover a remarkably simple counterexample to the Jacobian conjecture, a famous problem that has resisted mathematician...

Years ago, an audacious Fields medalist outlined a sweeping program that, he claimed, could be used to resolve a major problem in algebraic geometry. Other mathematicians had their doubts. Now he says...