Browsing by Author "Urzua, Giancarlo"

Now showing 1 - 13 of 13
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    ANTIFLIPS, MUTATIONS, AND UNBOUNDED SYMPLECTIC EMBEDDINGS OF RATIONAL HOMOLOGY BALLS
    (2021) Evans, Jonathan D.; Urzua, Giancarlo
    The Milnor fibre of a Q-Gorenstein smoothing of a Wahl singularity is a rational homology ball B-p,B-q. For a canonically polarised surface of general type X, it is known that there are bounds on the number p for which B-p,B-q admits a symplectic embedding into X. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded B-p,B-q into surfaces of general type equipped with non-canonical symplectic forms. Ultimately, these symplectic embeddings come from Mori's theory of flips, but we give an interpretation in terms of almost tonic structures and mutations of polygons. The key point is that a flip of surfacns, as studied by Hacking, Tevelev and Urmia, can be formulated as a combination of mutations of an almost toric structure and deformation of the symplectic form.
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    Arrangements of rational sections over curves and the varieties they define
    (EUROPEAN MATHEMATICAL SOC, 2011) Urzua, Giancarlo
    We introduce arrangements of rational sections over curves. They generalize line arrangements on P-2. Each arrangement of d sections defines a single curve in Pd-2 through the Kapranov's construction of (M) over bar (0,d+1). We show a one-to-one correspondence between arrangements of d sections and irreducible curves in M-0,M-d+1, giving also correspondences for two distinguished subclasses: transversal and simple crossing. Then, we associate to each arrangement A (and so to each irreducible curve in M-0,M-d+1) several families of nonsingular projective surfaces X of general type with Chern numbers asymptotically proportional to various log Chern numbers defined by A. For example, for the main families and over C, any such X is of positive index and pi(1)(X) similar or equal to pi 1 (A), where A is the normalization of A. In this way, any rational curve in M-0,M-d+1 produces simply connected surfaces with Chem numbers ratio bigger than 2. Inequalities like these come from log Chern inequalities, which are in general connected to geometric height inequalities (see Appendix). Along the way, we show examples of etale simply connected surfaces of general type in any characteristic violating any sort of Miyaoka Yau inequality.
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    Bounding Tangencies of Sections on Elliptic Surfaces
    (2021) Ulmer, Douglas; Urzua, Giancarlo
    Given an elliptic surface epsilon -> C over a field k of characteristic zero equipped with zero section O and another section P of infinite order, we give a simple and explicit upper bound on the number of points where O is tangent to a multiple of P.
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    Chilean configuration of conics, lines and points
    (2022) Dolgachev, Igor; Laface, Antonio; Persson, Ulf; Urzua, Giancarlo
    Using the theory of rational elliptic fibrations, we construct and discuss a one parameter family of configurations of 12 conics and 9 points in the projective plane that realizes an abstract configuration (12(6), 9(8)). This is analogous to the famous Hesse configuration of 12 lines and 9 points forming an abstract configuration (12(3), 9(4)). We also show that any Halphen elliptic fibration of index 2 with four triangular singular fibers arises from such configuration of conics.
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    Exotic surfaces
    (2024) Reyes, Javier; Urzua, Giancarlo
    Although exotic blow-ups of the projective plane at n points have been constructed for every n >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document}, the only examples known by means of rational blowdowns satisfy n >= 5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 5$$\end{document}. It has been an intriguing problem whether it is possible to decrease n. In this paper, we construct the first exotic CP2#4CP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}{\mathbb {P}}<^>2 \# 4 \overline{{\mathbb {C}}{\mathbb {P}}<^>2}$$\end{document} with this technique. We also construct exotic 3CP2#b-CP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3{\mathbb {C}}{\mathbb {P}}<^>2 \# b<^>- \overline{{\mathbb {C}}{\mathbb {P}}<^>2}$$\end{document} for b-=9,8,7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b<^>-=9,8,7$$\end{document}. All of them are minimal and symplectic, as they are produced from projective surfaces W with Wahl singularities and KW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_W$$\end{document} big and nef. In more generality, we elaborate on the problem of finding exotic (2 chi(OW)-1)CP2#(10 chi(OW)-KW2-1)CP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (2\chi ({\mathcal {O}}_W)-1) {\mathbb {C}}{\mathbb {P}}<^>2 \# (10\chi ({\mathcal {O}}_W)-K<^>2_W-1) \overline{{\mathbb {C}}{\mathbb {P}}<^>2} \end{aligned}$$\end{document}from these Koll & aacute;r-Shepherd-Barron-Alexeev surfaces W, obtaining explicit geometric obstructions on the corresponding configurations of rational curves.
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    Families of explicit quasi-hyperbolic and hyperbolic surfaces
    (2020) Garcia-Fritz, Natalia; Urzua, Giancarlo
    We construct explicit families of quasi-hyperbolic and hyperbolic surfaces parametrized by quasi-projective bases. The method we develop in this paper extends earlier works of Vojta and the first author for smooth surfaces to the case of singular surfaces, through the use of ramification indices on exceptional divisors. The novelty of the method allows us to obtain new results for the surface of cuboids, the generalized surfaces of cuboids, and other explicit families of Diophantine surfaces of general type. In particular, we produce new families of smooth complete intersection surfaces of multidegrees m1, horizontal ellipsis. These families give evidence for [6, Conjecture 0.18] in the case of surfaces.
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    On degenerations of Z/2-Godeaux surfaces
    (2022) Dias, Eduardo; Rito, Carlos; Urzua, Giancarlo
    We compute equations for the Coughlan's family of Godeaux surfaces with torsion Z/2, which we call Z/2-Godeaux surfaces, and we show that it is (at most) 7 dimensional. We classify all non-rational KSBA degenerations W of Z/2-Godeaux surfaces with one Wahl singularity, showing that W is birational to particular either Enriques surfaces, or D-2,D-n elliptic surfaces, with n = 3, 4 or 6. We present examples for all possibilities in the first case, and for n = 3, 4 in the second.
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    On the geography of line arrangements
    (2022) Eterovic, Sebastian; Figueroa, Fernando; Urzua, Giancarlo
    We present various results about the combinatorial properties of line arrangements in terms of the Chern numbers of the corresponding log surfaces. This resembles the study of the geography of surfaces of general type. We prove some new results about the distribution of Chern slopes, we show a connection between their accumulation points and the accumulation points of linear H-constants on the plane, and we conclude with two open problems in relation to geography over DOUBLE-STRUCK CAPITAL Q and over DOUBLE-STRUCK CAPITAL C.
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    On wormholes in the moduli space of surfaces
    (2022) Urzua, Giancarlo; Vilches, Nicolas
    We study a certain wormholing phenomenon that takes place in the Kollar-Shepherd-Barron-Alexeev (KSBA) compactification of the moduli space of surfaces of general type. It occurs because of the appearance of particular extremal P-resolutions in surfaces on the KSBA boundary. We state a general wormhole conjecture, and we prove it for a wide range of cases. At the end, we discuss some topological properties and open questions.
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    Rational configurations in K3 surfaces and simply-connected pg=1 surfaces with K2=1, 2, 3, 4, 5, 6, 7, 8, 9
    (2022) Reyes, Javier; Urzua, Giancarlo
    The aim of this paper is to directly relate the problem of constructing simply-connected p(g) = 1 surfaces of general type via Q-Gorenstein smoothings with geographical problems about arrangements of rational curves in K3 surfaces. We focus on simple normal crossing arrangements, showing numerical constraints based on their log Chern numbers and their rank in the Neron-Severi group. We also show what to expect as Wahl singularities, classifying the case when K-2 = 1. We use these arrangements to prove the existence of many (20 - 2K(2))-dimensional families of simply-connected surfaces with ample canonical class, p(g) = 1, and 1 <= K-2 <= 9, through several constructions with various special properties. We work out all our surfaces from one particular arrangement in a specific K3 surface with maximal Picard number. Our surfaces with K-2 = 7 and K-2 = 9 are the first surfaces known in the literature, together with the existence of a 4-dimensional family for K-2 = 8. With this, unobstructed simply-connected p(g) = 1 surfaces have been found for every possible K-2.
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    RIGID SURFACES ARBITRARILY CLOSE TO THE BOGOMOLOV-MIYAOKA-YAU LINE
    (2022) Stover, Matthew; Urzua, Giancarlo
    We prove the existence of rigid compact complex surfaces of general type whose Chern slopes are arbitrarily close to the Bogomolov-Miyaoka-Yau bound of 3. In addition, each of these surfaces has first Betti number equal to 4.
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    Simple embeddings of rational homology balls and antiflips
    (GEOMETRY & TOPOLOGY PUBLICATIONS, 2021) Park, Heesang; Shin, Dongsoo; Urzua, Giancarlo
    Let V be a regular neighborhood of a negative chain of 2-spheres (ie an exceptional divisor of a cyclic quotient singularity), and let B-p,B-q be a rational homology ball which is smoothly embedded in V. Assume that the embedding is simple, ie the corresponding rational blowup can be obtained by just a sequence of ordinary blowups from V. Then we show that this simple embedding comes from the semistable minimal model program (MMP) for 3-dimensional complex algebraic varieties under certain mild conditions. That is, one can find all simply embedded B-p,B-q 's in V via a finite sequence of antiflips applied to a trivial family over a disk. As applications, simple embeddings are impossible for chains of 2-spheres with self-intersections equal to. We also show that there are (infinitely many) pairs of disjoint Bp;q 's smoothly embedded in regular neighborhoods of (almost all) negative chains of 2-spheres. Along the way, we describe how MMP gives (infinitely many) pairs of disjoint rational homology balls Bp;q embedded in blown-up rational homology balls B-n,B-a (sic) CP2 (via certain divisorial contractions), and in the Milnor fibers of certain cyclic quotient surface singularities. This generalizes results of Khodorovskiy (2012, 2014), H Park, J Park and D Shin (2016) and Owens (2018) by means of a uniform point of view.
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    Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces
    (2022) Ulmer, Douglas; Urzua, Giancarlo
    We consider elliptic surfaces epsilon over a field k equipped with zero section O and another section P of infinite order. If k has characteristic zero, we show there are only finitely many points where O is tangent to a multiple of P. Equivalently, there is a finite list of integers such that if n is not divisible by any of them, then nP is not tangent to O. Such tangencies can be interpreted as unlikely intersections. If k has characteristic zero or p > 3 and epsilon is very general, then we show there are no tangencies between O and nP. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with K ample and K-2 unbounded.