On curves of genus eight

Curves without plane model of small degree

“…Let C denote a smooth irreducible projective curve of genus g > 0 defined over the field of complex numbers and let s C (2) be the minimal degree of plane models of C, i.e., the smallest degree of simple nets (birationally very ample linear series of dimension 2) on C. A simple g 2 sC (2) on C is clearly a complete and base point free linear series, and we have g ≤ (sC (2)−1)(sC (2)−2) 2 which implies the lower bound 3+ √ 8g+1 2 for s C (2). It is classically known ([12, Anhang G, §10]) that s C (2) = 2(g+4) 3 if C is a general curve of genus g, and we will show that any integer between these two numbers is attained by s C (2), for some curve C of genus g (Proposition 2.2).…”

“…It seems interesting to (dis-)prove that the claim of Proposition 3.1 remains true if g ≥ 3(t + 1) (i.e., ρ g (g + 1 − t, 2) ≥ 0); note that this genus bound just means that s C (2) > s 0 (2) := 2(g+4) 3 , the generic value.…”

“…Ifg = 0, we obtain, by (3.2), that gon(C) ≤ 6 but we can be a little bit more precise here. Assume that gon(C) ≥ 5; then, in (3.1), we can replace n − 4 by n − 5 since dim W 1 n (C) ≤ n − 5 according to [3,Theorem 2]. It follows n ≤ 5 and so gon(C) = 5.…”

Irreducibility and components rigid in moduli of the Hilbert scheme of smooth curves

“…Upon fixing a line L ⊂ P 3 , we consider the linear system D = P(H 0 (P 3 , I L (3))) consisting of cubics containing the line L. Note that any 4 given points on L impose independent conditions on cubics and hence dim D = dim P(H 0 (P 3 , O(3))) − 4 = 19 − 4 = 15. Since our curve C is completely determined by a pencil of cubics containing a line L ⊂ P 3 , we see that H 8,7,3 is a G(1, 15) bundle over G (1,3), the space of lines in P 3 . Hence H 8,7,3 is irreducible of dimension dim G(1, 15) + dim G(1, 3) = 28 + 4 = 32 = 4 · 8.…”

Very ample linear systems on blowings-up at general points of smooth projective varieties

“…So we assume for a general such L such a point P does not exist (in terms of linear systems: We obtain a base point free linear system g 1 5 ). In those cases C is birationally equivalent to a plane curve of degree 6 (for g = 10 see [16]; for g = 11 see [5]; for g = 8 see [2]). The associated map from C to P 2 with image that plane curve of degree 6 defines an invertible sheaf L belonging to W 2 6 (C) and again we find Cliff (C) ≤ 2.…”