Ragni Piene

• Piene, Ragni; Riener, Cordian & Shapiro, Boris (2024). Return of the evolute. Annales de l'Institut Fourier. ISSN 0373-0956. Full text in Research Archive Show summary

  Below we consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree d ≥ 2, we provide lower bounds for the following four numerical invariants: 1) the maximal number of times a real line can intersect the evolute of a real-algebraic curve of degree d; 2) the maximal number of real cusps which can occur on the evolute of a real-algebraic curve of degree d; 3) the maximal number of (cru)nodes which can occur on the dual curve to the evolute of a real-algebraic curve of degree d; 4) the maximal number of (cru)nodes which can occur on the evolute of a real-algebraic curve of degree d.

• Dickenstein, Alicia & Piene, Ragni (2023). Interpolation of toric varieties. arXiv.org. ISSN 2331-8422. Show summary

  Let $X\subset \mathbb P^d$ be a $m$-dimensional variety in $d$-dimensional projective space. Let $k$ be a positive integer such that $\binom{m+k}k \le d$. Consider the following \emph{interpolation problem}: does there exist a variety $Y\subset \mathbb P^d$ of dimension $\le \binom{m+k}k -1$, with $X\subset Y$, such that the tangent space to $Y$ at a point $p\in X$ is equal to the $k$th osculating space to $X$ at $p$, for almost all points $p\in X$? In this paper we consider this question in the \emph{toric} setting. We prove that if $X$ is toric, then there is a unique toric variety $Y$ solving the above interpolation problem. We identify $Y$ in the general case and we explicitly compute some of its invariants when $X$ is a toric curve.

• Piene, Ragni (2022). Higher order polar and reciprocal polar loci. In Aluffi, Paolo; Anderson, David; Hering, Milena; Mustata, Mircea & Payne, Sam (Ed.), Facets of Algebraic Geometry - A Collection in Honor of William Fulton's 80th Birthday, Volume 2. Cambridge University Press. ISSN 9781108792516. p. 238–253. Full text in Research Archive Show summary

  In this note we introduce higher order polar loci as natural generalizations of the classical polar loci, replacing the role of tangent spaces by that of higher order osculating spaces. The close connection between polar loci and dual varieties carries over to a connection between higher order polar loci and higher order dual varieties. We generalize the duality between the degrees of polar classes of a variety and those of its dual variety to varieties that are reflexive to a higher order. In particular, the degree of the top (highest codimension) polar class of order k is equal to the degree of the k-th dual variety. We define higher order Euclidean normal bundles and use them to define higher order reciprocal polar loci and classes. We give examples of how to compute the degrees of the higher order polar and reciprocal polar classes in some special cases: curves, scrolls, and toric varieties.

• Kleiman, Steven & Piene, Ragni (2022). Node Polynomials for Curves on Surfaces. SIGMA. Symmetry, Integrability and Geometry. ISSN 1815-0659. 18. doi: 10.3842/SIGMA.2022.059. Full text in Research Archive Show summary

  We complete the proof of a theorem we announced and partly proved in [Math. Nachr., vol. 271 (2004), Thm. 2.5, p. 74]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in that paper. It describes a natural cycle that enumerates the curves in the family with precisely r ordinary nodes. The second part is proved here. It asserts that, for r≤8, the class of this cycle is given by a computable universal polynomial in the pushdowns to the parameter space of products of the Chern classes of the family.

• Piene, Ragni; Kohn, Kathlén; Ranestad, Kristian; Rydell, Felix; Shapiro, Boris & Sinn, Rainer [Show all 8 contributors for this article] (2021). Adjoints and canonical forms of polypols. arXiv.org. ISSN 2331-8422. Show summary

  Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric perspective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numerator of the canonical differential form associated with the positive geometry. We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics

• Dickenstein, Alicia & Piene, Ragni (2017). Higher order selfdual toric varieties. Annali di Matematica Pura ed Applicata. ISSN 0373-3114. 196(5), p. 1759–1777. doi: 10.1007/s10231-017-0637-4. Full text in Research Archive Show summary

  The notion of higher order dual varieties of a projective variety, introduced in \cite{P83}, is a natural generalization of the classical notion of projective duality. In this paper we present geometric and combinatorial characterizations of those equivariant projective toric embeddings that satisfy higher order selfduality. We also give several examples and general constructions. In particular, we highlight the relation with Cayley-Bacharach questions and with Cayley configurations.

• Piene, Ragni (2016). Chern–Mather classes of toric varieties. arXiv.org. ISSN 2331-8422. Show summary

  The purpose of this short note is to prove a formula for the Chern--Mather classes of a toric variety in terms of its orbits and the local Euler obstructions at general points of each orbit (Theorem 2). We use the general definition of the Chern--Schwartz--MacPherson classes (see \cite{MR0361141}) and their special expression in case of a toric variety (see \cite{MR1197235}). As a corollary, we obtain a formula by Matsui--Takeuchi \cite[Corollary 1.6]{MR2737807}. Alternatively, one could deduce the formula of Theorem \ref{CM} from the Matsui--Takeuchi formula, by using our general result \cite[Th\'eor\`eme 3]{MR1074588} for the degree of the polar varieties in terms of the Chern--Mather classes.

• Lanteri, Antonio; Mallavibarrena, Raquel & Piene, Ragni (2015). Inflectional loci of quadric fibrations. Journal of Algebra. ISSN 0021-8693. 441, p. 363–397. doi: 10.1016/j.jalgebra.2015.06.023. Full text in Research Archive Show summary

  Quadric fibrations over smooth curves are investigated with respect to their osculatory behavior. In particular, bounds for the dimensions of the osculating spaces are determined, and explicit formulas for the classes of the inflectional loci are exhibited under appropriate assumptions. Moreover, a precise description of the inflectional loci is provided in several cases. The associated projective bundle and its image in the ambient projective space of the quadric fibration, the enveloping ruled variety, play a significant role. Several examples are discussed to illustrate concretely the various situations arising in the analysis.

• Piene, Ragni (2015). Discriminants, polytopes, and toric geometry. In Cartier, Pierre; Choudary, A. D. R. & Waldschmidt, Michel (Ed.), Mathematics in the 21st Century. Springer Science+Business Media B.V.. ISSN 978-3-0348-0858-3. p. 151–162. doi: 10.1007/978-3-0348-0859-0_9. Show summary

  Toric varieties can be considered as a meeting point of algebra, geometry and combinatorics. Though they constitute only a small subset of all algebraic varieties, their structure is rich enough to make them interesting when formulating and testing conjectures. Because of their combinatorial description, results in algebraic geometry can some times be proved using combinatorics, and combinatorial results can sometimes be proved by algebraic-geometric methods. In this note we will survey results by several authors concerning the description of dual and higher order dual varieties of toric varieties in terms of their associated lattice polytopes. Toric varieties that are ruled correspond to Cayley polytopes, and we will describe several properties and characterizations of Cayley polytopes using toric geometry.

• Piene, Ragni (2015). Polar varieties revisited. In Gutierrez, Jaime; Schicho, Josef & Weimann, Martin (Ed.), Computer Algebra and Polynomials. Springer Science+Business Media B.V.. ISSN 978-3-319-15080-2. p. 139–150. doi: 10.1007/978-3-319-15081-9_8. Show summary

  We recall the definition of classical polar varieties, as well as those of affine and projective reciprocal polar varieties. The latter are defined with respect to a non-degenerate quadric, which gives us a notion of orthogonality. In particular we relate the reciprocal polar varieties to the ``Euclidean geometry'' in projective space. The Euclidean distance degree and the degree of the focal loci can be expressed in terms of the ranks, i.e., the degrees of the classical polar varieties, and hence these characters can be found also for singular varieties, when one can express the ranks in terms of the singularities.

• Piene, Ragni (2014). Algebraic spline geometry: some remarks. In Dokken, Tor & Muntingh, Agnar Georg Peder (Ed.), SAGA – Advances in ShApes, Geometry, and Algebra. Results from the Marie Curie Initial Training Network. Springer. ISSN 978-3-319-08634-7. p. 169–175. doi: 10.1007/978-3-319-08635-4__9.
• Dickenstein, Alicia; Di Rocco, Sandra & Piene, Ragni (2014). Higher order duality and toric embeddings. Annales de l'Institut Fourier. ISSN 0373-0956. 64(1), p. 375–400. doi: 10.5802/aif.2851. Show summary

  ABSTRACT. The notion of higher order dual varieties of a projective variety, introduced by Piene in 1983, is a natural generalization of the classical notion of projective duality. In this paper we study higher order dual varieties of projective toric embeddings. We compute the degree of the second dual variety of a 2-jet spanned projective embedding of a smooth toric threefold in geometric and combinatorial terms, and we classify those whose second dual variety has dimension less than expected. We also describe the tropicalization of the k-th dual variety of an equivariantly embedded (not necessarily normal) toric variety.

• Lanteri, Antonio; Mallavibarrena, Raquel & Piene, Ragni (2012). Inflectional Loci of Scrolls over Smooth, Projective Varieties. Indiana University Mathematics Journal. ISSN 0022-2518. 61(2), p. 717–750. doi: 10.1512/iumj.2012.61.4630.
• Kleiman, Steven; Piene, Ragni & Tyomkin, Ilya (2011). Enriques diagrams, arbitrarily near points, and Hilbert schemes. Rendiconti Lincei - Matematica e Applicazioni. ISSN 1120-6330. 22(4), p. 411–451. doi: 10.4171/RLM/608.
• Dickenstein, A; Di Rocco, S & Piene, Ragni (2009). Classifying smooth lattice polytopes via toric fibrations. Advances in Mathematics. ISSN 0001-8708. 222(1), p. 240–254. doi: 10.1016/j.aim.2009.04.002.
• Johansen, Pål Hermunn; Piene, Ragni & Løberg, Magnus Bjørnsen (2008). Monoid hypersurfaces. In Jüttler, Bert & Piene, Ragni (Ed.), Geometric Modeling and Algebraic Geometry. Springer. ISSN 978-3-540-72184-0. p. 55–77.
• Mork, Heidi Camilla & Piene, Ragni (2008). Polars of real singular plane curves. In Dickenstein, Alicia; Schreyer, Frank Olaf & Sommese, Andrew J. (Ed.), Algorithms in Algebraic Geometry. Springer Science+Business Media B.V.. ISSN 978-0-387-75154-2. p. 99–115.
• Lanteri, Antonio; Mallavibarrena, Raquel & Piene, Ragni (2008). Inflectional loci of scrolls. Mathematische Zeitschrift. ISSN 0025-5874. 258, p. 557–564. doi: 10.1007/s00209-007-0185-5.

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• Holden, Helge & Piene, Ragni (2019). The Abel Prize 2013-2017. Springer Nature. ISBN 978-3-319-99027-9. 774 p.
• Holden, Helge & Piene, Ragni (2014). The Abel Prize 2008–2012. Springer Science+Business Media B.V.. ISBN 978-3-642-39448-5. 571 p.
• Holden, Helge & Piene, Ragni (2010). The Abel Prize 2003-2007. The First Five Years. Springer Publishing Company. ISBN 978-3-642-01372-0. 327 p.
• Jüttler, Bert & Piene, Ragni (2008). Geometric Modeling and Algebraic Geometry. Springer. ISBN 978-3-540-72184-0. 232 p.
• Piene, Ragni; Elkadi, Mohamed & Mourrain, Bernard (2006). Algebraic geometry and geometric modeling. Springer. ISBN 978-3-540-33274-9. 252 p.

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• Piene, Ragni (2024). Envelopes and evolutes of curves and surfaces. Show summary

  Evolutes of curves and surfaces were studied in the 19th century (Monge, Darboux, …) and in the 20th century (Porteous, …) in the setting of affine real differential geometry. Salmon, in the 19th century, considered the case of complex projective curves and surfaces. In this talk I will define evolutes og a variety as “iterated” envelopes of the family of “Euclidean normal spaces” to the variety. Using the theory of singularities of mappings, Thom polynomials yield numerical formulas. These formulas agree with those of Salmon in the cases considered by him.

• Piene, Ragni (2023). Curve counting and generating functions. Show summary

  A classical problem in enumerative geometry is to determine how many plane curves of given degree and with given types of singularities pass through some fixed points. For example, there is 1 non-singular cubic passing through 9 points, but 12 singular cubics passing through 8 points. Such curve counting problems have received new attention because of their appearance in string theory in theoretical physics. There is currently no closed formula solution to the above general problem. However, as I will explain, in some cases one can determine the generating function: a formal power series whose coefficients are the desired solutions.

• Piene, Ragni & Shapiro, Boris (2023). Return of the plane evolute. Show summary

  We consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree d ≥ 2, we provide lower bounds for the following four numerical invariants: 1) the maximal number of times a real line can intersect the evolute of a real-algebraic curve of degree d; 2) the maximal number of real cusps which can occur on the evolute of a real-algebraic curve of degree d; 3) the maximal number of (cru)nodes which can occur on the dual curve to the evolute of a real-algebraic curve of degree d; 4) the maximal number of (cru)nodes which can occur on the evolute of a real-algebraic curve of degree d.

• Piene, Ragni (2023). Interpolation of toric varieties. Show summary

  We address the following interpolation problem: given a variety X in projective n-space, does there exist a variety Y in the same space, containing X, and such that the tangent space to Y at almost any point of X is equal to the kth osculating space to X at that point? We give a positive answer in the case that both X and Y are toric varieties. The special case where X is the rational normal curve is studied, in particular for k=2 and k=n-1.

• Piene, Ragni (2022). Positive geometries: their adjoints and canonical forms. Show summary

  In recent work by physicists, positive geometries are defined as certain semi-algebraic sets together with a meromorphic differential form called the canonical form: calculating scattering amplitudes reduces to determining the canonical form. Examples of planar positive geometries are polygons and, more generally, “curved” polygons, so-called polypols. The latter were proposed by Wachspress as generalized algebraic finite elements. In his quest to define barycentric coordinates for polypols, he introduced the adjoint curve of a rational polypol. We show that a rational regular polypol gives a positive geometry and give an explicit expression for its canonical form in terms of the adjoint and boundary curves of the polypol. In the special case that the polypol is a convex polygon, we show that the adjoint curve is hyperbolic and describe its nested ovals.    This talk is based on joint work with K. Kohn, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn,  M.-S. Sorea, and S. Telen.

• Piene, Ragni (2022). Planar polypols, their adjoints and canonical forms. Show summary

  Planar polypols - “polygons with curved sides” - were proposed by Eugene Wachspress as generalized algebraic finite elements. In order to define barycentric coordinates for polypols, he introduced the adjoint curve of a rational polypol. In recent work by physicists, positive geometries are defined as certain semi-algebraic sets together with a meromorphic differential form called the canonical form. We show that a rational regular polypol gives a positive geometry and give an explicit expression for its canonical form in terms of the adjoint and boundary curves of the polypol. In the special case that the polypol is a convex polygon, we show that the adjoint curve is hyperbolic and describe its nested ovals.    This talk is based on joint work with K. Kohn, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn,  M.-S. Sorea, and S. Telen.

• Piene, Ragni (2022). Long paper's journey into arXiv: “Node polynomials for curves on surfaces”.
• Piene, Ragni (2021). Envelopes of plane curves: return of the evolute. Show summary

  The evolute of a curve in the Euclidean plane is the locus of its centers of curvature. It can also be viewed as the envelope of its normals, and the study of the evolutes of conics goes back to Apollonius. Evolutes and related curves were studied by the geometers of the 19th century, as witnessed by Salmon’s book (1852) on “higher plane curves”. Recently there has been considerable interest in the Euclidean distance degree and discriminant of algebraic varieties. For a plane curve, this discriminant is the same as the evolute. Though much is known about evolutes, there are also many open problems which merit further investigations. In this talk, I will first consider the case of curves in the projective complex plane. I will define dual and reciprocal curves, envelopes, and evolutes, and give rigorous proofs of some classical enumerative results. In the case of real curves much less is known. The problems I will consider concern how many of the various singular points of the evolute, corresponding to e.g. the vertices and diameters of the given curve, can be real. I will give partial answers and illustrate with examples. This talk is based on joint work with Cordian Riener and Boris Shapiro.

• Piene, Ragni (2021). Singular curves on a moving surface. Show summary

  A classical problem in enumerativ geometry is to determine the number of singular curves in a given linear system on a smooth projective surface, for example the number of curves with r nodes in an r-dimensional linear system. Göttsche's conjecture (proved by Tzeng and Kool–Shende–Thomas) says that these numbers are obtained by evaluating certain universal polynomials in the Chern numbers of the system and the surface. For various reasons, it is interesting to consider the similar problem for r-nodal curves on a family of surfaces. Continuing the work of I. Vainsencher, S. Kleiman and I conjectured – and proved for r<9 – that there exist universal polynomials also in this case, and that they have a certain shape. The first part of this conjecture has now been proved by T. Laarakker. In this talk I will give an argument, inspired by work of N. Qviller, that gives evidence for the conjectured shape of these polynomials. Ingredients are Bell polynomials, configuration spaces, polydiagonals, and residual intersection theory.

• Riener, Cordian & Piene, Ragni (2020). Ragni Piene. In Paycha, Sylvie & Matoff, Tovia (Ed.), Women of mathematics throughout Europe - A gallery of portraits. Verlag am Fluss. ISSN 978-3-9814032-7-5. p. 122–129.
• Piene, Ragni (2020). Partitions, polynomials and generating functions. Show summary

  The solution to a counting problem can be a closed formula or a generating function: the number of ways to triangulate a polygon with $n+2$ vertices is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$, whereas the number of partitions of an integer $n$ is given as the coefficient of $q^n$ in the power series expansion of the partition function $P(q)=\prod_m (1-q^m)^{-1}$. The partition function and its generalization the MacMahon function, as well as the generating functions for the Catalan and the Bell numbers, turn up as solutions to various counting problems in algebraic geometry, for example in Schubert calculus and in Gromow--Witten and Donaldson--Thomas theory. In the talk I will give some examples and some hints at explanations.

• Piene, Ragni (2019). Polydiagonals and Bell polynomials in curve counting. Show summary

  Given a family of curves in a family of surfaces, consider the class representing the r-nodal curves. Recently Ties Laarakker showed that this class is given by a universal polynomial in pushdowns of products of Chern classes of the family. This proves the first part of a conjecture by Kleiman-Piene; the second part says that these polynomials are Bell polynomials. I will explain how Laarakker’s result can be used to show the second part, by generalizing work of Qviller.

• Piene, Ragni (2019). Projective geometry from a toric point of view. Show summary

  Classical projective geometry addresses questions like classification, duality, divisors, sections and projections, enumerative geometry, etc. Although projective toric varieties form but a small subset of all projective varieties, they constitute nonetheless a rich and interesting playground for the study of these questions. The fact that there is a "dictionary" between toric projective varieties and convex lattice polytopes makes it possible to use combinatorial methods to prove algebraic geometrical results, and vice versa. In the talk, I will give several examples of such results.

• Piene, Ragni (2019). Projective geometry from a toric point of view. Show summary

  The study of curves, surfaces, and higher dimensional varieties in projective space involves questions about tangency, duality, divisors, sections and projections, enumerative geometry, and classification. Although projective toric varieties form but a small subset of all projective varieties, they constitute nonetheless a rich and interesting playground for the study of these questions. The fact that there is a ``dictionary'' between projective toric varieties and convex lattice polytopes makes it possible to use combinatorial methods to prove algebraic geometric results, and vice versa. In the talk, I will give several examples of such results.

• Piene, Ragni (2019). Discriminants and polytopes in toric geometry. Show summary

  Toric varieties play an important role at the crossroad of algebra, geometry and combinatorics. Toric geometry is both rigid and rich, and allows for testing conjectures and proving results in algebraic geometry using combinatorial methods, and in combinatorics using algebro-geometrical methods. An example of the latter is a combinatorial characterization of certain Cayley polytopes. This talk will survey recent work by various authors concerning the description and characterization of dual and higher order dual varieties of projective toric varieties in terms of the corresponding lattice configurations and polytopes.

• Piene, Ragni (2019). Higher order polar and reciprocal polar varieties. Show summary

  Let $X\subset \mathbb P^N$ be a projective variety of dimension $m$. For $x\in X$, there is a sequence of osculating spaces to $X$ at $x$: \[\{x\}\subseteq T_x \subseteq \Osc_x^2 \subseteq \Osc_x^3\subseteq \cdots \subseteq \mathbb P^N.\] If $\dim \Osc_x^k < N$ for a general point $x\in X$, then the \emph{$k$th dual variety} $X^{(k)}\subset (\mathbb P^N)^\vee$ of $X$ is the set of hyperplanes containing a $k$th osculating space to $X$. The \emph{higher order polar varieties} of $X$ are obtained by imposing Schubert conditions on the osculating spaces, similar to the classical case. The \emph{higher reciprocal polar varieties} are obtained by imposing Schubert conditions on the ``Euclidean'' normal spaces to the osculating spaces. I will give examples of this theory, with special emphasis on toric varieties.

• Piene, Ragni (2019). Counting problems and generating functions. Show summary

  To a sequence of integers $a_1,a_2,a_3,\ldots$ is associated a generating function: the formal power series $f(x)=\sum a_n x^n$. The generating function provides a way of displaying the sequence. For example, if $a_n$ denotes the number of ways one can write the integer $n$ as a sum of positive integers, the generating function is the partition function $p(x)=\Pi _{m\ge 1}(1 - x^m)^{-1}$, and hence one can compute $a_n$ as the coefficient of $x_n$ in the power series expansion of this function. The partition function and its generalization the MacMahon function, as well as the generating functions for the Catalan and the Bell numbers, turn up in various geometric settings --- triangulations, Schubert calculus, Gromow--Witten and Donaldson--Thomas invariants. In the talk I will give some examples and some hints at explanations. https://www.youtube.com/watch?v=CkdNUhcl5cM&list=PLo4jXE-LdDTSBTsTWp4NL-qAdz7g74gKW&index=2

• Lanteri, Antonio; Mallavibarrena, Raquel & Piene, Ragni (2018). Corrigendum to “Inflectional loci of quadric fibrations” [J. Algebra 441 (2015) 363–397] (S0021869315003221) (10.1016/j.jalgebra.2015.06.023)). Journal of Algebra. ISSN 0021-8693. 508, p. 589–591. doi: 10.1016/j.jalgebra.2018.04.012.
• Piene, Ragni (2018). Reciprocal polar varieties. Show summary

  The polar varieties of a projective variety are defined via Schubert conditions on the tangent spaces. By introducing “Euclidean normal spaces”, one can define “reciprocal polar varieties” by imposing conditions on the normal spaces. The degrees of the reciprocal polar varieties are sums of degrees of polar varieties, and the degree of the “top” reciprocal polar variety is the so called Euclidean distance degree introduced by Sturmfels et al, which has received recent attention in applied real algebraic geometry.​

• Persson, Ulf & Piene, Ragni (2018). Interview with Ragni Piene. [Journal]. EMS Newsletter.
• Piene, Ragni (2017). Euclidean projective geometry, reciprocal polar varieties, and focal loci. Show summary

  We can regard an affine space as projective space minus a hyperplane at infinity, together with a notion of perpendicularity in that hyperplane. This gives a “Euclidean” structure on the affine space, including a Euclidean normal bundle of a given variety. This is used to define the reciprocal polar varieties, the end point map, and the focal loci. The degrees of the reciprocal polar varieties can be expressed in terms of the degrees of the ordinary polar varieties, also in the case that the variety is singular. There has been recent interest in computing, or bounding, these degrees. I will give several examples, especially in the case of curves, surfaces, and toric varieties.

• Piene, Ragni (2017). Polar varieties and Euclidean distance degree. Show summary

  Reciprocal polar varieties of a given, possibly singular, projective variety are defined with respect to a non-degenerate quadric hypersurface. The quadric induces a notion of orthogonality in the projective space, hence also a notion of ``Euclidean geometry''. Thus one can define the Euclidean distance degree and the Euclidean normal bundle, as well as the classical concepts of focal loci and caustics of reflections. The Euclidean distance degree and the degree of the focal locus can be expressed in terms of the degrees of the classical polar varieties. I will give some examples in the case of curves, surfaces, and toric varieties.

• Piene, Ragni (2017). Algebraic geometry for geometric modeling. Show summary

  To model an object, one can try to get an approximate the shape by putting together simple geometric objects like line segments and plane patches. To allow for curved shapes, one needs different objects -- the simplest ones being pieces of conics or spheres or cylinders, so-called natural quadrics. With increased computer hardware and software, it is possible to go further and use objects defined by polynomials of higher degree.

• Piene, Ragni (2017). Euclidean projective geometry: reciprocal polar varieties and focal loci. Show summary

  Euclidean orthogonality and ``distance'' can be defined in a given (real or complex) projective space with respect to a quadric, or a quadric in the hyperplane at infinity. Using this, one defines the Euclidean normal bundle and reciprocal polar varieties of a given, possibly singular, projective variety. These polar varieties are related to the classical concepts of focal loci and caustics of reflections. In fact, the focal locus is the branch locus of the end point map of the Euclidean normal bundle. There has been recent interest in computing, or bounding, the degree of the focal locus. I will give examples of such computations, especially in the case of curves, surfaces, and toric varieties.

• Piene, Ragni (2016). Algebraic splines and generalized Stanley–Reisner rings. Show summary

  Given a simplicial complex $\Delta\subset \mathbb R^d$, let $C^r_k(\Delta)$ denote the vector space of piecewise polynomial functions (algebraic splines) of degree $\le k$ and smoothness $r$. A major problem is to determine the dimension (and construct bases) of these vector spaces. Pioneering work by Billera, Rose, Schenck, and others gave upper and lower bounds using homological methods. The ring of continuous splines $C^0(\Delta)=C^0_k(\Delta)$ is (essentially) equal to the face ring, or Stanley--Reisner ring, of $\Delta$ and has the property that its geometric realization describes $\Delta$. More precisely, the part of ${\rm Spec}(C^0(\Delta))$ lying in a certain hyperplane and having nonnegative coordinates is ``equal'' to $\Delta$. Here we shall consider the \emph{generalized Stanley--Reisner rings} $C^r(\Delta):=\oplus_k C^r_k(\Delta)\subset C^0(\Delta)$. We present a conjectural description of ${\rm Spec}(C^r(\Delta))$ generalizing the one for $r=0$. To illustrate the conjecture, some very simple examples will be given.

• Piene, Ragni (2016). Enumeration of singular curves on surfaces. Show summary

  The number of $r$-nodal curves in a fixed linear system on a smooth surface passing through the appropriate number of points is given by a polynomial in the Chern numbers of the surface. There are various generalizations of this enumerative question, e.g. to algebraic systems of curves on algebraic families of surfaces, or to curves with other singularities than nodes. Another generalization is to consider the case of singular surfaces. One could hope that one could obtain the corresponding formulas by replacing the Chern classes of the smooth surface with the Chern--Mather classes of the singular surface. Since for toric surfaces the Chern--Mather classes are known -- they are equal to the Chern--Schwartz--MacPherson classes weighted by the local Euler obstruction -- toric surfaces provide a good testing ground. However, as shown by Liu--Osserman, who considered some special toric surfaces, the situation is not that simple. In the talk I will discuss these problems, as well as some of the different approaches to solving them and understanding their generating functions.

• Piene, Ragni (2016). Algebraic splines and generalized Stanley–Reisner rings. Show summary

  Given a simplicial complex $\Delta\subset \mathbb R^d$, let $C^r_k(\Delta)$ denote the vector space of piecewise polynomial functions (algebraic splines) of degree $\le k$ and smoothness $r$. A major problem is to determine the dimension of these vector spaces. Pioneering work by Billera, Rose, Schenck, and others give upper and lower bounds using homological methods. Here we shall consider the rings $C^r(\Delta):=\oplus_k C^r_k(\Delta)$ that we shall call the generalized Stanley--Reisner rings of $\Delta$. The ring of continuous splines $C^0(\Delta)$ is (essentially) the face ring of $\Delta$ and has the property that its geometric realization describes $\Delta$. More precisely, the part of ${\rm Spec}(C^0(\Delta)$ lying in a certain hyperplane and having nonnegative coordinates ``is'' $\Delta$. I propose a conjectural generalization of this situation, giving a description of ${\rm Spec}(C^r(\Delta))$ for $r\ge 0$. To support the conjecture, some very simple examples will be given.

• Piene, Ragni (2015). Projective geometry from a toric point of view. Show summary

  Classical projective geometry addresses questions like classification, duality, divisors, sections and projections, enumerative geometry, etc. Although projective \emph{toric} varieties form but a small subset of all projective varieties, they constitute nonetheless a rich and interesting playground for the study of such questions. The fact that there is a ``dictionary'' between toric projective varieties and lattice point configurations and convex lattice polytopes, makes it possible to use combinatorial methods to prove algebraic geometric results, and vice versa. In this talk I will survey some recent results by various authors, especially concerning the classification of ``hollow'' polytopes and the characterization of selfdual and higher order selfdual toric varieties. In both cases, so-called Cayley polytopes play a major role.

• Piene, Ragni (2015). Counting curves on singular surfaces. Show summary

  Let $S$ be a smooth projective surface and $\mathcal L$ a line bundle. As is now well known, the number of $r$-nodal curves in the linear system $|\mathcal L|$ passing through he appropriate number of points on $S$ can be expressed as a polynomial of degree $r$ in the Chern numbers $\mathcal L^2$, $K_S\cdot \mathcal L$, $K_S^2$, and $c_2(S)$. There has recently been works by several authors (Ardila--Block, Liu--Osserman, Block--G\"ottsche) that attempt to find similar formulas in the case that $S$ is a singular toric surface. I will discuss this work, and also initial recent work by N\o dland in the case of weighted projective planes.

• Piene, Ragni (2015). Higher order self-dual toric varieties. Show summary

  To a projective variety one associates its dual variety: the set of hyperplanes tangent to the given variety. It is well known that the only smooth varieties that are ``self-dual'', i.e., are such that the dual variety is isomorphic to the variety, are quadric hypersurfaces in characteristic 0 and Fermat hypersurfaces in characteristic $p>0$. By replacing tangency conditions by higher order contact conditions, one can define \emph{higher order} dual varieties and ask for a classification of those that are ``higher'' self-dual. In this talk I will report on joint work with Alicia Dickenstein, where we study higher order dual varieties of \emph{toric embeddings}. A lattice point configuration $\mathcal A \subset \mathbb Z^n$ defines a (real or complex) toric embedding in $\mathbb P^N$, where $N=\# \mathcal A -1$. The aim is to characterize those varieties that are isomorphic to one of its higher order dual varieties, in particular to find conditions on the configuration $\mathcal A$ or its convex lattice polytope $P={\rm Conv}(\mathcal A)$ for this to happen. I will explain our results, give examples, and state some conjectures.

• Piene, Ragni (2014). Higher order osculating spaces and dual varieties of toric varieties. Show summary

  The dual varieties of a projective variety can be studied via the conormal varieties lying in the point-hyperplane incidence variety. In the case the variety is a toric embedding, properties of the dual varieties can be translated into properties of the lattice point configuration or polytope. In particular, one can give combinatorial criteria for such a variety to be dual defective or (higher) selfdual. Several examples, including Cayley configurations and rational normal scrolls, will be given. Connections with diophantine problems will be highlighted. This is joint work with Alicia Dickenstein.

• Piene, Ragni (2014). An interview with Cheryl Praeger. Seoul Intelligencer. p. 16–18.
• Jemterud, Torkild & Piene, Ragni (2014). Ekko. [Radio]. NRK.
• Piene, Ragni & Næss Olsen, Maren (2014). Mysteriet med kvinnene som forsvant. [Newspaper]. Morgenbladet.
• Piene, Ragni & Sundlisæter, Tale (2013). Profilen: Ragni Piene. [Business/trade/industry journal]. Teknisk Ukeblad.
• Piene, Ragni (2013). Polar varieties revisited. Show summary

  Polar varieties and polar classes have played an important role in the study and classification of projective varieties. In particular they were used to give geometric definitions of Todd classes and Chern classes. Their local counterpart was used in singularity theory. More recently they have been studied and generalized also in the context of real projective and affine geometry, with applications to finding points on real components, to compute Euclidean distance degrees, to caustics by reflection.

• Piene, Ragni (2013). Node polynomials for curves on surfaces. Show summary

  Consider a family of smooth projective surfaces F over Y and a relative divisor D on F. A generalized version of Göttsche's conjecture (proved by Tzeng and Kool–Shende–Thomas in the case of a linear system on a fixed surface) says that the class in Y representing fibres that are r-nodal can be expressed as a Bell polynomial in the relative Chern classes of D and F. We explain the reason for this conjecture and sketch the proof of it in the case r less than or equal to 8.

• Piene, Ragni (2013). Enriques diagrams and equisingular strata of families of curves. Show summary

  Given a singular point on a curve lying on a smooth surface one associates a weighted sequence of infinitely near points, combinatorially expressed by an Enriques diagram. The singularity defines a fat point on the surface, and the Enriques diagram can be recovered also from the fat point. For a given Enriques diagram and a given surface, the set of fat points with this diagram forms a smooth subscheme of the Hilbert scheme of the surface. The dimension of the subscheme is expressed in terms of the numerical invariants of the diagram, and I will relate the latter to the numerical invariants of the singularity. I will briefly mention how this theory can be applied to study curves in a family with singularities of prescribed topological type. This is joint work with Steven Kleiman.

• Piene, Ragni (2013). Polytopes, discriminants and toric geometry. Show summary

  This talk will survey recent work by various authors concerning the description of dual and higher order dual varieties of projective toric varieties and the corresponding lattice configurations and polytopes. The interplay between algebraic geometry and combinatorics makes it possible to prove algebro-geometric statements using combinatorial methods, and combinatorial statements using algebro-geometrical methods. An example of the latter is a combinatorial characterization of certain Cayley polytopes.

• Piene, Ragni (2013). Discriminants, polytopes and toric geometry. Show summary

  Toric varieties play an important role at the crossroad of algebra, geometry and combinatorics. Toric geometry is both rigid and rich, and allows for testing conjectures and proving results in algebraic geometry using combinatorial methods, and in combinatorics using algebro-geometrical methods. An example of the latter is a combinatorial characterization of certain Cayley polytopes. This talk will survey recent work by various authors concerning the description and characterization of dual and higher order dual varieties of projective toric varieties in terms of the corresponding lattice configurations and polytopes. In particular we will characterize lattice configurations which give self dual, or higher self dual, toric varieties.

• Piene, Ragni (2013). Higher order selfdual toric varieties. Show summary

  A lattice point configuration A in Z^n defines a (real or complex) toric embedding in P^N, where N=#A -1. We want to characterize those varieties that are isomorphic to one of its higher order dual varieties, in particular find conditions on the configuration A or its convex lattice polytope P=Conv(A) for this to happen. This is joint work with Alicia Dickenstein, and generalizes previous work by Bourel, Dickenstein, Rittatore in the case of ordinary self dual toric varieties.

• Piene, Ragni (2012). Goldbachs formodning. [Radio]. "Det du tror, men ikke kan bevise", Radioselskapet, NRK.
• Piene, Ragni (2012). Abelprisen 2012. [Radio]. Dagsnytt atten - NRK.
• Piene, Ragni (2012). Cayley polytopes and toric geometry. Show summary

  The “Cayley trick” expresses the resultant of two polynomials in one variable as the discriminant of a polynomial in two variables. Generalized and translated into the language of lattice polytopes and toric geometry, the trick consists in building a n-dimensional polytope from k + 1 polytopes of dimension n − k so that the new polytope has no interior lattice points. Such a polytope is called a Cayley polytope. Let P ⊂ R^n be a convex lattice polytope. The codegree of P is an integer between 1 and n + 1 that measures the “hollowness” of P : it is the smallest integer m such that the dilated polytope mP contains interior lattice points. A Cayley polytope as above has codegree at least k + 1. A recent result, due to Dickenstein, Di Rocco, Nill, and Piene, says that the only polytopes of dimension n and codegree at least (n + 3)/2 are Cayley polytopes with k at least (n + 1)/2. The proof relies on the study of the polarized toric variety (X, L) defined by P .

• Holden, Helge & Piene, Ragni (2012). Matematikkens gave. Aftenposten (morgenutg. : trykt utg.). ISSN 0804-3116.
• Piene, Ragni (2012). The Abel Prize – the first 10 years.
• Piene, Ragni (2011). Enumerative geometry related to mirror symmetry and SYZ.
• Piene, Ragni (2011). Abelprisen til banebryter. [Newspaper]. Aftenposten.
• Piene, Ragni (2011). Matematyka jest wszechobecna. Wiadomości Matematyczne. 47(2), p. 221–223.
• Piene, Ragni (2011). Higher order dual varieties: the toric case. Show summary

  Given a projective variety $X\subset \PP^m$, the dual variety $X^{(1)} \subset {\PP^m}^\vee$ is the set of hyperplanes tangent to $X$. More generally, the $k$th dual variety $X^{(k)}$ is the set of hyperplanes tangent to $X$ to order $k$, i.e., containing a $k$th osculating space to $X$. A variety is called $k$-defective if the dimension of $X^{(k)}$ is less than expected. In the case that the embedding $X\subset \PP^m$ is toric, with associated polytope $P$, the degree of $X^{(1)}$ can be expressed in terms of the lattice volumes of the faces of $P$. We show that there is a similar formula for the degree of the $2$-dual of a toric threefold, and we use this expression to show that the only $2$-jet spanned toric embedding that is $2$-defective is $(\PP^3,\mathcal O_{\PP^3}(2))$, i.e., the toric embedding with associated polytope $2\Delta_3$. This is joint work with Alicia Dickenstein and Sandra Di Rocco.

• Piene, Ragni (2011). Higher order dual varieties - the toric case. Show summary

  The notion of higher order dual varieties is a natural generalization of the classical notion of dual varieties. They are defined by using higher order osculating spaces instead of tangent spaces. In the case of toric varieties, the degree of the higher order dual varieties can be expressed in combinatorial terms. We shall in particular consider the case of rational normal scrolls, and the case of toric threefolds (joint work with A. Dickenstein and S. Di Rocco).

• Piene, Ragni (2011). The problematic art of counting. Show summary

  Many counting problems, like \begin{itemize} \item[] ``In how many ways can a positive integer $n$ be written as a sum of positive integers?'' \item[] ``Given a polytope $P$, how many lattice points does the dilated polytope $nP$ contain?'' \item[] ``How many lines in a $(n+1)$-dimensional space meet $2n$ general $(n-1) $-planes?'' \end{itemize} are solved by finding a closed form for the corresponding \emph{generating function} $\sum_n N_nq^n$, where the $N_n$ are the sought numbers and $q$ is a variable. In this lecture we shall, in addition to the above questions, also address an old problem from enumerative geometry: \begin{itemize} \item[] ``How many plane curves of degree $d$ have $r$ singularities and pass through $\frac{d(d+3)}{2} -r$ given points in the plane?'' \end{itemize} In this case the generating function is still unknown, but there has recently been substantial progress on the problem and its generalizations.

• Piene, Ragni (2011). Generating functions in enumerative geometry. Show summary

  To a sequence of integers $a_1,a_2,a_3,\ldots$ is associated a generating function: the formal power series $f(x)=\sum a_n x^n$. The generating function provides a way of displaying the sequence. For example, if $a_n$ denotes the number of ways one can write the integer $n$ as a sum of positive integers, the generating function is the partition function $p(x)=\Pi _{m\ge 1}(1 - x^m)^{-1}$, and hence one can compute $a_n$ as the coefficient of $x_n$ in the power series expansion of this function. The partition function and its generalization the MacMahon function, as well as the generating functions for the Catalan and the Bell numbers, turn up in recent problems of counting certain geometric objects, such as Gromov--Witten and Donaldson--Thomas invariants. In the talk I will give some examples and some hints at explanations.

• Piene, Ragni (2011). Generating functions and enumerative geometry.
• D'Andrea, Carlos; Giusti, Marc; Pardo, Luis M. & Piene, Ragni (2010). Effective methods in algebraic geometry 2009: Barcelona. Guest editors' foreword. Journal of symbolic computation. ISSN 0747-7171. 45(12), p. 1251–1253. doi: 10.1016/j.jsc.2010.06.007.
• Piene, Ragni (2010). Mathematics ought to be a subject very suited for women. [Newspaper]. ICWM Newsletter.
• Piene, Ragni (2010). And who chooses the winners? [Internet]. http://plus.maths.org/content/and-who-chooses-winners. Show summary

  What's the point of the Fields Medal and other maths prizes? Who decides who gets one? And when will we have the first female medallist? Rachel talks to László Lovász, current president of the International Mathematical Union (IMU), Martin Grötschel, the IMU's secretary, and Ragni Piene, the new chair of the Abel Prize committee, about all this and more.

• Piene, Ragni (2010). Some Counting Problems and Their Generating Functions. Show summary

  To a sequence of integers a_1,a_2,a_3, ... is associated with a generating function: the formal power series ƒ(x)=∑a_nX_n. The generating function provides a way of displaying the sequence. For example,if an denotes the number of ways one can write the integer n as a sum of positive integers, the generating function is the partition function p(x)=Π_{m≥1}(1 - x^m)^{-1}, and hence one can compute an as the coefficient of xn in the power series expansion of this function. The partition function and its generalization the MacMahon function, as well as the generating functions for the Catalan and the Bell numbers, turn up in recent problems of counting certain geometric objects (like Gromov-Witten and Donaldson-Thomas invariants). Without going into details, the talk will give some examples and some hints at explanations.

• Piene, Ragni (2010). Inflection loci of projective varieties. Show summary

  Rational normal curves are the only smooth projective curves that have no inflection points (of any order). In fact this property characterizes Veronese embeddings of projective spaces of any dimension. For a given class of varieties, e.g. toric varieties or scrolls, one can ask to characterize those that are uninflected. In some cases it is possible to compute the degree of the inflection locus, and hence give criteria for it to be empty. The talk will survey some old and new results, and focus on recent joint work with A. Lanteri and R. Mallavibarrena concerning scrolls.

• Piene, Ragni (2010). Classification des polytopes entiers par fibrations toriques. Show summary

  Un polytope entier $P$ est un polytope dans $\mathbb R^n$ avec sommets dans $\mathbb Z^n$. Un tel polytope peut ou non contenir des points entiers intérieurs. Si non, son double $2P$, ou triple $3P$, ou ... veut en contenir - on appelle le codegre de $P$ le plus petit entier $m$ tel que $mP$ contient des points entiers intérieurs. Le but est de classifier les polytopes tels que m soit grand par rapport à n, ce qu'on achève par des méthodes des variétés dites toriques associées aux polytopes.

• Piene, Ragni (2010). Curve counting and generating functions. Show summary

  Generating functions are a main ingredient in enumerative combinatorics. In enumerative algebraic geometry, however, they were only introduced around 1990, when physicists were able to predict the generating function of the number of rational curves on certain Calabi–Yau threefolds by using the principle of mirror symmetry. Since then, the hunting for generating functions of other curve counting problems has intensified. The talk will explain some successes and some failures in this hunt, with particular focus on singular curves on surfaces.

• Piene, Ragni (2009). Generating functions in enumerative geometry. Show summary

  Abstract. I will explain some problems in enumerative algebraic geometry concerning the counting of curves. In certain cases, solutions exist in the form of explicit generating functions, in other cases there are only conjectures as to the shape of these functions. In particular, I will consider the case of nodal curves on an algebraic surface (joint work with Steve Kleiman).

• Piene, Ragni (2009). Functors of infinitely near points on an algebraic surface and the counting of Gromov-Witten invariants.
• Piene, Ragni (2009). Counting curves: the hunting of generating functions. Show summary

  Generating functions are a main ingredient in enumerative combinatorics. In enumerative algebraic geometry, however, they were only introduced around 1990, when physicists surprisingly were able to predict the generating function of rational curves on Calabi-Yau threefolds by using the principle of mirror symmetry. Since then, the hunting for generating functions for other curve counting problems has intensified. Ragni Piene will explain some successes and some failures in this hunt, with particular focus on curves on surfaces.

• Piene, Ragni (2008). Classifying regular lattice polytopes via toric fibrations.
• Piene, Ragni (2008). Infinitely near points, Enriques diagrams, and Hilbert schemes.
• Piene, Ragni (2008). Classifying smooth lattice polytopes via toric fibrations.
• Piene, Ragni (2008). Classifying lattice polytopes using toric fibrations.
• Piene, Ragni (2008). Classifying regular lattice polytopes using toric fibrations.
• Piene, Ragni (2007). Equal to Nobel. [Newspaper]. Frontline, Vol. 24, Issue 07 (India).
• Piene, Ragni (2007). Real monoid surfaces. Show summary

  A monoid surface is a surface of degree d which has a singular point of multiplicity d-1. Any monoid surface admits a rational parameterization, hence is of potential interest in computer aided geometric design. The possible real forms of the singularities on a monoid surface are determined. These results are applied to the classification of quartic monoid surfaces and a study of a stratification of the parameter space of these surfaces. A part of this work is joint with M. Løberg and P.H. Johansen, the other part is due to P.H. Johansen.

• Piene, Ragni (2007). Curves and surfaces. Show summary

  This talk is intended as a brief introduction to certain aspects of the theory of algebraic curves and surfaces. To a projective algebraic variety one associates its arithmetic genus, using the constant term of the Hilbert polynomial. Since a complex projective algebraic curve can be viewed as a compact Riemann surface, it also has a topological genus, equal to the number of holes in the Riemann surface. Hirzebruch's Riemann-Roch theorem says that the topological genus is equal to the arithmetic genus. I sketch a proof of this fundamental result, using algebra, geometry, and topology. For algebraic surfaces there is a corresponding result, Noether's formula, expressing the arithmetic genus in terms of topological invariants, which can be proved in a similar way. The last part of the talk discusses the theory of curves on surfaces, in particular the links to classical enumerative geometry and to modern string theory in theoretical physics.

• Piene, Ragni (2007). Inflectional loci of scrolls.
• Piene, Ragni (2006). Polar and dual varieties of real curves and surfaces. Show summary

  The theory of polar varieties have been used by Bank et al. and Safey El Din et al. to give algorithms for finding a point on each connected component of a smooth, complete intersection real variety. In this talk, based on ongoing joint work with Heidi Mork, we shall consider possibly singular varieties and study the polar varieties and their relation to the dual variety and to the Gauss map. In particular, we shall look at the case of curves and surfaces.

• Piene, Ragni (2006). Enumerating singular curves on a surface. Show summary

  Given a family of curves on an algebraic surface and a set of curve singularity types, there is a natural cycle representing curves of the family that have such singularities. The goal is to compute these cycle classes, or even to determine their generating functions. I shall report on joint work with S. Kleiman on this problem.

• Moe, Torgunn Karoline; Piene, Ragni & Ranestad, Kristian (2013). Cuspidal curves on Hirzebruch surfaces. Akademika forlag.
• Muntingh, Agnar Georg Peder; Piene, Ragni & Winther, Ragnar (2011). Topics in Polynomial Interpolation Theory. Unipub forlag. ISSN 1501-7710. 1056(1056).

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