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Register a new userIntersection multiplicity of a sparse curve and a low-degree curve
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Added by
Mateusz Skomra
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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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 that the multiplicity of any isolated solution $(a,b) in mathbb{C}^2$ with nonzero coordinates is no greater than $frac{5}{2}d^2t^2$. We ask whether this intersection multiplicity can be polynomially bounded in the number of monomials of $F$ and $G$, and we briefly review some connections between sparse polynomials and algebraic complexity theory.
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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.
Let $C$ be a smooth projective curve of genus $2$. Following a method by O Grady, we construct a semismall desingularization $tilde{mathcal{M}}_{Dol}^G$ of the moduli space $mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,mathbb{C}), SL(2,mathbb{C})$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of $tilde{mathcal{M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $mathcal{M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $mathcal{M}_{Dol}^G$ and prove that the mixed Hodge structure on it is actually pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.
In this paper we consider the Brill-Noether locus $W_{underline d}(C)$ of line bundles of multidegree $underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $ggeq2$. We give an explicit description of the irreducible components of $W_{underline d}(C)$ for a semistable multidegre $underline d$. As a consequence we show that, if two semistable multidegrees of total degre $g-1$ on a curve with no rational components differ by a twister, then the respective Brill-Noether loci have isomorphic components.
We continue our study of genus 2 curves $C$ that admit a cover $ C to E$ to a genus 1 curve $E$ of prime degree $n$. These curves $C$ form an irreducible 2-dimensional subvariety $L_n$ of the moduli space $M_2$ of genus 2 curves. Here we study the case $n=5$. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general $n$. We compute a normal form for the curves in the locus $L_5$ and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of $L_5$ as subvarieties of $M_2$ and classify all curves in these loci which have extra automorphisms.
We explore the cohomological structure for the (possibly singular) moduli of $mathrm{SL}_n$-Higgs bundles for arbitrary degree on a genus g curve with respect to an effective divisor of degree >2g-2. We prove a support theorem for the $mathrm{SL}_n$-Hitchin fibration extending de Cataldos support theorem in the nonsingular case, and a version of the Hausel-Thaddeus topological mirror symmetry conjecture for intersection cohomology. This implies a generalization of the Harder-Narasimhan theorem concerning semistable vector bundles for any degree. Our main tool is an Ng^{o}-type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.
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