Canonical geometrically ruled surfaces

Abstract

We prove the existence of canonical scrolls; that is, scrolls playing the role of canonical curves. First of all, they provide the geometrical version of Riemann Roch Theorem: any special scroll is the projection of a canonical scroll and they allow to understand the classification of special scrolls in P N . Canonical scrolls correspond to the projective model of canonical geometrically ruled surfaces over a smooth curve. We also prove that the generic canonical scroll is projectively normal except in the hyperelliptic case and for very particular cases in the nonhyperelliptic situation.

Jacobian discrepancies and rational singularities

“…Let C be a non-hyperelliptic curve of genus g. Fix a non-special divisor B on C such that deg B ≥ 2g + 1 (note that linear system |2(B − K C )| is very ample and thus contains smooth elements), and let E = O C ⊕ ω C (−B). Following [FP05], the ruled surface π : S = P C (E) → C is a canonical geometrically ruled surface. Moreover, if H = C 0 + π * B where C 0 is the section determined by the quotient E → O C , then the linear system |H| determines a projectively normal embedding of S as a canonical scroll in some P N (see Theorem 6.16 of [FP05]).…”

“…Following [FP05], the ruled surface π : S = P C (E) → C is a canonical geometrically ruled surface. Moreover, if H = C 0 + π * B where C 0 is the section determined by the quotient E → O C , then the linear system |H| determines a projectively normal embedding of S as a canonical scroll in some P N (see Theorem 6.16 of [FP05]). Let X be the cone over S ⊂ P N .…”

The Generic Special Scroll of Genusgin PN: Special Scrolls in P3

“…Proof: Because R is special it has a special curve X a (see [4]). This is the projection of a canonical curve.…”

Brill–Noether theory and non-special scrolls

“…For further details on ruled surfaces, we refer to [19], [21, § V], [2], [9], [10], [11], [12], [15], [16], [24], [25], [26], [30] and [32]. If we denote by H the section of ρ such that…”