2025/03/26 by Davide Franco, Alessandra Sarti, Franco, Davide +1
Computer Science · Mathematics · #14B05 #14E15 #14F43 #14F45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2503.20958
openalex publication_date 2025/03/26 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
We give two explicit versions of the decomposition theorem of Beilinson, Bernstein and Deligne applied to the universal family of quartic surfaces of ℙ3. The starting point of our investigation is the remark that the nodes of a quartic surface impose independent conditions to the linear system | O\mathbb P3(4)|. Although this property is known in literature, we provide a different argument more suited to our purposes. By a result of \citeDGF, the independence of the nodes implies in turn that each component of Severi's variety is smooth of the expected dimension and that the dual variety is a divisor with normal crossings around Severi's variety. This allows us to study the complex Rπ_*ℚX, the derived direct image of the constant sheaf over the universal family of quartic surfaces X \stackrelπ\longrightarrow \mathbb P34, both in the open set parametrizing smooth and nodal quartics and in a tubular neighborhood of the variety of Kummer surfaces. We obtain in both cases an explicit decomposition and a formality result for the complex Rπ_*ℚX.