The regular splice diagram determines the embedded topology of a generic ﬁbre and the degree of each horizontal curve.
Rational polynomials of simple type
The degree of a horizontal curve E is the degree of the restriction ˜f |E .
Rational polynomials of simple type
Russell (correctly presented in ) constructed an example of a rational polynomial with no degree one horizontal curves.
Rational polynomials of simple type
Moreover, if φ is a closed curve on L then −Ω([φ]) is the degree of the map κ(φ− ).
Hamiltonian symplectomorphisms and the Berry phase
Fqi ’s are a chain of rational curves connecting C and Γ and they will be contracted under stable reduction; the only nontrivial part is Γ ⊂ Sα which maps to E with a degree m map.
A Simple Proof that Rational Curves on K3 are Nodal
Among these are the moduli space of quartic curves (equivalently, of degree two Del Pezzo surfaces) and of rational elliptic ﬁbrations admitting a section (equivalently, of degree one Del Pezzo surfaces).
Compactifications defined by arrangements I: the ball quotient case
We now assume that (T , W ) is an adjoint torus of type E7 . A smooth quartic curve Q in P2 determines a Del Pezzo surface of degree 2: the double cover S → P2 ramiﬁed along Q.
Compactifications defined by arrangements I: the ball quotient case
It will turn out, that in the case minimal degree complete pencils of a general canonical curve C ⊂ Pg−1 all these correspondences are the same.
Geometric Syzygies of Mukai Varieties and General Canonical Curves with Genus at most 8
The “curved” galaxies we sought were identiﬁed as those whose sextupole moment was oriented so that one of its minima was aligned within a few degrees of a quadrupole minimum.
Observations of cluster substructure using weakly lensed sextupole moments
The splitting type k .(C ) of NC , which we call the normal type of C , is a natural global numerical invariant of the embedding C ⊂ Pn , perhaps the most fundamental such invariant beyond the degree, and thus the problem of enumerating curves with given normal type seems a natural one.
Normal bundles of rational curves in projective spaces
Note that putting together Theorem 6.1, Corollaries 6.2-6.4 and Example 5.5 we now know the splitting type of the normal bundle of a generic rational curve or rational angle of every degree and bidegree.
Normal bundles of rational curves in projective spaces
As a result, the locus of these rational angles will appear as an improper part of the locus of curves of degree d = a + b with unbalanced normal bundle.
Normal bundles of rational curves in projective spaces
Therefore W is relatively very ample in a neighborhood of X0 and embeds a general ﬁbre Xs as a smooth rational curve of degree d in Pn , thus providing an explicit smoothing of Ca,b .
Normal bundles of rational curves in projective spaces
Appendix, i.e. a smooth model of a generic 1-parameter family of rational curves of degree d in Pn incident to a generic collection (A.) of linear spaces.
Normal bundles of rational curves in projective spaces
The result of Flenner gives a bound for the degree of the curve and the result of Bogomolov shows moreover that the restriction is in fact semistable for every smooth curve fulﬁlling a stronger degree condition.
Looking out for stable syzygy bundles
***