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The tangent cone to a convex surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t0921201.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t0921202.png" /> is the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t0921203.png" /> of the cone formed by the half-lines emanating from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t0921204.png" /> and intersecting the [[Convex body|convex body]] bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t0921205.png" /> in at least one point distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t0921206.png" />. (This cone itself is sometimes called the solid tangent cone.) In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t0921207.png" /> is the boundary of the intersection of all half-spaces containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t0921208.png" /> and defined by the supporting planes to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t0921209.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212011.png" /> is a plane, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212012.png" /> is called a smooth point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212013.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212014.png" /> is a dihedral angle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212015.png" /> is called a ridge point; finally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212016.png" /> is a non-degenerate (convex) cone, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212017.png" /> is called a conic point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212018.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian) {{MR|0346714}} {{MR|0244909}} {{ZBL|0311.53067}} </TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser (1979) pp. 13–59 {{MR|0568493}} {{ZBL|0427.52003}} </TD></TR></table>

The tangent cone to an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212019.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212020.png" /> is the set of limiting positions of the secants passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212021.png" />. More precisely, if the algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212022.png" /> is imbedded in an affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212023.png" /> and if it is defined by an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212024.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212025.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212026.png" /> has coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212027.png" />, then the tangent cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212028.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212030.png" /> is given by the ideal of initial forms of the polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212031.png" />. (If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212032.png" /> is the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212033.png" /> in homogeneous polynomials and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212035.png" /> is called the initial form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212036.png" />.) There is another definition, suitable for Noetherian schemes (see [[#References|[1]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212037.png" /> be the [[Local ring|local ring]] of a [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212038.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212039.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212040.png" /> be its maximal ideal. Then the spectrum of the graded ring

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212041.png" /></td> </tr></table>

is called the tangent cone to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212042.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212043.png" />.

In a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212044.png" /> the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212045.png" /> is, in a certain sense, structured in the same way as the tangent cone. For example, if the tangent cone is reduced, normal or regular, then so is the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212046.png" />. The dimension and multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212047.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212048.png" /> are the same as the dimension of the tangent cone and the multiplicity at its vertex. The tangent cone coincides with the [[Zariski tangent space|Zariski tangent space]] if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212049.png" /> is a non-singular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212050.png" />. A morphism of varieties induces a mapping of the tangent cones.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-i. Igusa, "Normal point and tangent cone of an algebraic variety" ''Mem. Coll. Sci. Univ. Kyoto'' , '''27''' (1952) pp. 189–201 {{MR|0052155}} {{ZBL|0101.38501}} {{ZBL|0049.38504}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1967) {{MR|0213347}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero I, II" ''Ann. of Math.'' , '''79''' (1964) pp. 109–203; 205–326 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Whitney, "Local properties of analytic varieties" S.S. Cairns (ed.) , ''Differential and Combinatorial Topol. (Symp. in honor of M. Morse)'' , Princeton Univ. Press (1965) pp. 205–244 {{MR|0188486}} {{ZBL|0129.39402}} </TD></TR></table>

''V.I. Danilov''

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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>