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W3211760425 abstract "The family of complex projective surfaces in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper P cubed> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {P}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> having precisely <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ</mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> nodes as their only singularities has codimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ</mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the linear system <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue script upper O Subscript double-struck upper P cubed Baseline left-parenthesis d right-parenthesis EndAbsoluteValue> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>|{mathcal O}_{mathbb {P}^3}(d)|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for sufficiently large <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and is of degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Subscript delta comma double-struck upper C Superscript double-struck upper P cubed Baseline left-parenthesis d right-parenthesis equals left-parenthesis 4 left-parenthesis d minus 1 right-parenthesis cubed right-parenthesis Superscript delta Baseline slash delta factorial plus upper O left-parenthesis d Superscript 3 delta minus 3 Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mn>3</mml:mn> </mml:msup> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>δ</mml:mi> </mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>!</mml:mo> <mml:mo>+</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>3</mml:mn> <mml:mi>δ</mml:mi> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>N_{delta ,mathbb {C}}^{mathbb {P}^3}(d)=(4(d-1)^3)^delta /delta !+O(d^{3delta -3})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Subscript delta comma double-struck upper C Superscript double-struck upper P cubed Baseline left-parenthesis d right-parenthesis> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>N_{delta ,mathbb {C}}^{mathbb {P}^3}(d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is polynomial in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By means of tropical geometry, we explicitly describe <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis 4 d cubed right-parenthesis Superscript delta Baseline slash delta factorial plus upper O left-parenthesis d Superscript 3 delta minus 1 Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>4</mml:mn> <mml:msup> <mml:mi>d</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>δ</mml:mi> </mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>!</mml:mo> <mml:mo>+</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>3</mml:mn> <mml:mi>δ</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(4d^3)^delta /delta !+O(d^{3delta -1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces passing through a suitable generic configuration of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n equals StartBinomialOrMatrix d plus 3 Choose 3 EndBinomialOrMatrix minus delta minus 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-OPEN> <mml:mo maxsize=1.2em minsize=1.2em>(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mfrac linethickness=0> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-CLOSE> <mml:mo maxsize=1.2em minsize=1.2em>)</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:mo>−</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n=binom {d+3}{3}-delta -1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper P cubed> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {P}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These surfaces are close to tropical limits which we characterize combinatorially, introducing the concept of floor plans for multinodal tropical surfaces. The concept of floor plans is similar to the well-known floor diagrams (a combinatorial tool for tropical curve counts): with it, we keep the combinatorial essentials of a multinodal tropical surface <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are sufficient to reconstruct <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the real case, we estimate the range for possible numbers of real multi-nodal surfaces satisfying point conditions. We show that, for a special configuration <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold-italic w> <mml:semantics> <mml:mi mathvariant=bold-italic>w</mml:mi> <mml:annotation encoding=application/x-tex>boldsymbol {w}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of real points, the number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Subscript delta comma double-struck upper R Superscript double-struck upper P cubed Baseline left-parenthesis d comma bold-italic w right-parenthesis> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant=bold-italic>w</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>N_{delta ,mathbb {R}}^{mathbb {P}^3}(d,boldsymbol {w})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of real surfaces of degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> having <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta> <mml:semantics> <mml:mi>δ</mml:mi> <mml:annotation encoding=application/x-tex>delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> real nodes and passing through <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold-italic w> <mml:semantics> <mml:mi mathvariant=bold-italic>w</mml:mi> <mml:annotation encoding=application/x-tex>boldsymbol {w}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded from below by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis three halves d cubed right-parenthesis Superscript delta Baseline slash delta factorial plus upper O left-parenthesis d Superscript 3 delta minus 1 Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msup> <mml:mi>d</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>δ</mml:mi> </mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>!</mml:mo> <mml:mo>+</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>3</mml:mn> <mml:mi>δ</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>left (frac {3}{2}d^3right )^delta /delta ! +O(d^{3delta -1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove analogous statements for counts of multinodal surfaces in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper P Superscript 1 Baseline times double-struck upper P squared> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {P}^1times mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper P Superscript 1 Baseline times double-struck upper P Superscript 1 Baseline times double-struck upper P Superscript 1> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {P}^1times mathbb {P}^1times mathbb {P}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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W3211760425 date "2021-11-02" @default.
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W3211760425 title "Tropical floor plans and enumeration of complex and real multi-nodal surfaces" @default.
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W3211760425 doi "https://doi.org/10.1090/jag/774" @default.
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