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We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let $N geq 2$, and consider an isolated complete intersection curve singularity germ $f colon (mathbb{C}^N,0) to (mathbb{C}^{N-1},0)$. We introduce a numerical function $m mapsto operatorname{AD}_{(2)}^m(f)$ that arises as an error term when counting $m^{mathrm{th}}$-order weight-$2$ inflection points with ramification sequence $(0, dots, 0, 2)$ in a $1$-parameter family of curves acquiring the singularity $f = 0$, and we compute $operatorname{AD}_{(2)}^m(f)$ for various $(f,m)$. Particularly, for a node defined by $f colon (x,y) mapsto xy$, we prove that $operatorname{AD}_{(2)}^m(xy) = {{m+1} choose 4},$ and we deduce as a corollary that $operatorname{AD}_{(2)}^m(f) geq (operatorname{mult}_0 Delta_f) cdot {{m+1} choose 4}$ for any $f$, where $operatorname{mult}_0 Delta_f$ is the multiplicity of the discriminant $Delta_f$ at the origin in the deformation space. Furthermore, we show that the function $m mapsto operatorname{AD}_{(2)}^m(f) -(operatorname{mult}_0 Delta_f) cdot {{m+1} choose 4}$ is an analytic invariant measuring how much the singularity counts as an inflection point. We obtain similar results for weight-$2$ inflection points with ramification sequence $(0, dots, 0, 1,1)$ and for weight-$1$ inflection points, and we apply our results to solve various related enumerative problems.
We generalize Werners defect formula for nodal hypersurfaces in $mathbb P^{4}$ to the case of a nodal complete intersection threefold.
We give an elementary combinatorial proof of the following fact: Every real or complex analytic complete intersection germ X is equisingular -- in the sense of the Hilbert-Samuel function -- with a germ of an algebraic set defined by sufficiently long truncations of the defining equations of X.
By the fundamental work of Griffiths one knows that, under suitable assumption, homological and algebraic equivalence do not coincide for a general hypersurface section of a smooth projective variety $Y$. In the present paper we prove the same result in case $Y$ has isolated singularities.
Let $F(x, y) in mathbb{C}[x,y]$ be a polynomial of degree $d$ and let $G(x,y) in mathbb{C}[x,y]$ be a polynomial with $t$ monomials. We want to estimate the maximal multiplicity of a solution of the system $F(x,y) = G(x,y) = 0$. Our main result is th
A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real morsifications for rea