Rational curves and rational singularities

 We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a ℂ^*-action. For varieties with an isolated singularity, covered by a family of rational curves with a general member not passing through the singular point, we show that this singularity is rational. In particular, this provides an explanation of classical results due to H. A. Schwartz and G. H. Halphen on polynomial solutions of the generalized Fermat equation. Received: 7 May 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 14J17, 14L30, 13H10     

On the singularities of polar curves

Let be a given equisingularity class of irreducible algebroid plane curves with just one characteristic exponent. We first obtain the equisingularity type of the general polar of a curve of with general moduli and we also determine the Newton-Cramer polygon of the general polar of all curves in .     

Baum-Bott indices for curves of singularities

Let F be a holomorphic foliation on Pⁿ by curves such that the components of its singular locus are curves C_i and points p_j. We compute the Baum-Bott indices BB_φ(F, C_i) in terms of the main invariants of F and C_i. We also determine the sum of the BB_φ(F, p_i) in terms of the same invariants.When φ corresponds to the determinant, the latter result generalizes, from special to all holomorphic foliations, a formula for the number of isolated singularities of F, counted with multiplicities.     

Foliations of isonergy surfaces and singularities of curves

It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level.We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.      

On the trees associated to plane curve singularities and ideals

Si comparano diverse nozioni d’equisingolarità per le curve tracciate sopra una superficie complessa liscia, e per gli ideali di dimensione finita dell’anello locale d’uno dei suoi punti. Si vede che, in generale, questi nozioni sono diverse, ma (per le curve) queste coincidono se esse fossero membri della stessa famiglia continua.     

On the ideals and singularities of secant varieties of Segre varieties

We find minimal generators for the ideals of secant varietiesof Segre varieties in the cases of _k(¹ x ⁿ x ^m) for all k, n,m, ₂(ⁿ x ^m x ^p x ^r) for all n, m, p, r (GSS conjecture for fourfactors), and ₃(ⁿ x ^m x ^p) for all n, m, p and prove they arenormal with rational singularities in the first case and arithmeticallyCohen–Macaulay in the second two cases.     

Uniquely generated Grothendieck space ideals

Starting from a result of Pietsch on the Grothendieck ideal of strictly nuclear locally convex spaces, we classify all those Grothendieck ideals of nuclear locally convex spaces that are generated by a unique operator ideal on the class of Banach spaces.     

Autoduality for treelike curves with planar singularities

Let C be a treelike curve, that is, a reduced curve whose irreducible components meet transversally at disconnecting points ofC. AssumeC has planar singularities. In this note we prove an Autoduality Theorem for C. More precisely, in [6] a natural Abel map A: C → $\bar J$ to a certain compactification $\bar J$ of the generalized Jacobian of C has been constructed. In this note we prove that the pullback map $A^* :Pic^0 (\bar J) \to Pic^0 (C)$ is an isomorphism.     

Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′²(t)+y′²(t)+z′²(t)≡σ²(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ³ is inherently more complicated than that of the corresponding set in ℝ². We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ³), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.