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A REFINEMENT OF THE CLASSICAL CLIFFORD INEQUALITY
Iliev, Hristo Korean Mathematical Society 2007 대한수학회지 Vol.44 No.3
We offer a refinement of the classical Clifford inequality about special linear series on smooth irreducible complex curves. Namely, we prove about curves of genus g and odd gonality at least 5 that for any linear series $g^r_d$ with $d{\leq}g+1$, the inequality $3r{\leq}d$ holds, except in a few sporadic cases. Further, we show that the dimension of the set of curves in the moduli space for which there exists a linear series $g^r_d$ with d<3r for $d{\leq}g+l,\;0{\leq}l{\leq}\frac{g}{2}-3$, is bounded by $2g-1+\frac{1}{3}(g+2l+1)$.
A refinement of the classical Clifford inequality
Hristo Iliev 대한수학회 2007 대한수학회지 Vol.44 No.3
We offer a refinement of the classical Clifford inequality aboutspecial linear series on smooth irreducible complex curves.Namely, we prove about curves of genus g and odd gonality atleast 5 that for any linear series g^r_d with d leq g+1, theinequality 3r leq d holds, except in a few sporadic cases.Further, we show that the dimension of the set of curves in themoduli space for which there exists a linear series g^r_d withd < 3r for d leq g+l, 0 leq l leq frac{g}{2}-3, isbounded by 2g-1 + frac{1}{3} (g+2l+1).
On the Irreducibility of the Hilbert Scheme of Curves in 5
Taylor Francis 2008 Communications in Algebra Vol.36 No.4
<P> Denote by Id,g,r the Hilbert scheme parametrizing smooth irreducible complex curves of degree d and genus g embedded in r. Severi (1921) claimed that Id,g,r is irreducible if d ≥ g + r. Ein proved that the conjecture is true for r = 3 and 4, and in general that Id,g,r is irreducible if [image omitted] (Ein, 1986, 1987). As it is known, for r ≥ 6 the conjecture is incorrect and r = 5 remains the only unsettled case. Here I prove that Id,g,5 is irreducible, if [image omitted], which doesn't yet resolve Severi's conjecture for r = 5, but expands the known irreducibility range.</P>