2025/01/03 by Alexandru Dimca, Dimca, Alexandru
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2501.01807
openalex publication_date 2025/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let C be a reduced complex projective plane curve, and let d1 and d2 be the first two smallest exponents of C. For a free curve C of degree d, there is a simple formula relating d,d1, d2 and the total Tjurina number of C. Our first result discusses how this result changes when the curve C is no longer free. For a free line arrangement, the Poincaré polynomial coincides with the Betti polynomial B(t) and with the product P(t)=(1+d1t)(1+d2t). Our second result shows that for any curve C, the difference P(t)-B(t) is a polynomial a t +bt2, with a and b non-negative integers. Moreover a =0 or b=0 if and only if C is a free line arrangement. Finally we give new bounds for the second exponent d2 of a line arrangement \mathcal A, the corresponding lower bound being an improvement of a result by H. Schenck concerning the relation between the maximal exponent of \mathcal A and the maximal multiplicity of points in \mathcal A.