Young Rock Kim

Star-Configurations in ℙn and the Weak-Lefschetz Property

We find that some union of two star-configurations in ℙ2 has generic Hilbert function. Applying the result, we prove that some Artinian quotients of a coordinate ring of a star-configuration in ℙn satisfy the weak-Lefschetz property. More precisely, let 𝕏 and 𝕐 be star-configurations in ℙ2 of type (2, s) and (2, s + 1) defined by forms F1,…, Fs, and G1,…, Gs, L, respectively, with deg(Fi) = deg(Gi) ≤2 for i = 1,…, s and s ≥ 3. If L is a general linear form in R = 𝕜[x0, x1, x2], then R/(I𝕏 + I𝕐) has the weak-Lefschetz property with a Lefschetz element L, which extends the result of [21].

NORMAL GENERATION OF NONSPECIAL LINE BUNDLES ON ALGEBRAIC CURVES

In this paper, we classify ( C;L) such that a smooth curve C of genus g has a nonspecial very ample line bundle L of deg L = 2 g 2 a failing to be normally generated, in terms of the number a.

QUARTET CONSISTENCY COUNT METHOD FOR RECONSTRUCTING PHYLOGENETIC TREES

Among the distance based algorithms in phylogenetic tree reconstruction, the neighbor-joining algorithm has been a widely used and eective method. We propose a new algorithm which counts the number of consistent quartets for cherry picking with tie breaking. We show that the success rate of the new algorithm is almost equal to that of neighbor-joining. This gives an explanation of the qualitative nature of neighbor-joining and that of dissimilarity maps from DNA sequence data. Moreover, the new algorithm always reconstructs correct trees from quartet consistent dissimilarity maps.

PROJECTIVE NORMALITY OF ALGEBRAIC CURVES AND ITS APPLICATION TO SURFACES

Let L be a very ample line bundle on a smooth curve C of genus g with (3g + 3) 2 deg L 2g 5. Then L is normally generated if deg L max 2g + 2 4h 1 (C, L), 2g (g 1) 6 2h 1 (C, L) . Let C be a triple covering of genus p curve C with C C and D a divisor on C with 4 p deg D (g 1) 6 2 p. Then KC( D) becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.